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Philosophical transactions of the
Royal Society of London
Royal Society (Great Britain), JSTOR (Orgariizatiori)
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R
PHILOSOPHICAL
TRANSACTIONS.
/ {^ ■' ' :i
OF THE
ROYAL SOCIETY OF LONDON.
Seiiies a.
CONTAINING PAPEES OF A MATHEMATICAL OK PHYSICAL CHAKACTEE.
VOL. 195.
LONDON:
IMJfNTED I5Y IIARKISON AM) SONS, ST. MARTINS LANB, W.O.,
)rinttTS in #rbinBni lo $ti Pajtsig.
January, 1901,
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C m ]
CONTENTS.
(A)
VOL. 195.
List of Illustrations page ▼
Advertisement vii
List of Institutions entitled to receive the Philosophical Transactions or Proceedings of the Royal
Society ' ix
Adjudication of Medals xvii
X T. Mathematical Coritribntions to the Theory of Evolution, — VIl. On the Con'elatio7i
of Characters not Quantitatively Measurable. By Karl Peaiison, F.Ii.S.
page 1
11. Electrical Conductivity in Gases Traversed by Cathode Rays. By J. C.
McLennan, Demonstrator in Physics^ University of Toronto. Communicated
by Professor J. J. Thomson, F.R.S. 49
^ 111. Mathematical Contnbutions to the Theory of Evolution. — VIII. On the Inheri
tance of Character's not capable of Exact Qvxintitative Meamrement. —
Part I. Introductory. Part II. On the Inheritance of Coatcolour in Horses.
Part III. On the Inheritance of Eyecolour in Man. By Kabl Peabson,
F.R.S. , ivith the assistance of Alice Lee, D.Sc.y University College,
London 79
IV. On Simultaneous Partial Differential Equations. By A, C. Dixon, Sc.D.
Communicoied by J. W. L. Glaibheu, Sc.D 151
a 2
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[ iv ]
V. The Velocity of the Ions produced in Gases hy Ronlgen Rays. By John Zeleny,
B.Sc.y B.A., Assistant Professor of Physics, University of Minnesota, Com
municated hy Professor J. J. Thomson, F.R.S. 193
VI. Underground Temperature at Oxford in the Yexxr 1899, as determined hy Five
Platinumresistance Thermometers. By Arthur A. Rambaut, M.A., D.Sc,
Radcliffe Ohserver. Communicated hy E. H. Griffiths, F.R.S. . . . 235
VII. The Diffusion of Ions produced in Air hy the Action of a Radioactive
SuhstancCy Ultraviolet Light and Point Discharges. By John S. Townsend,
M.A., ClerkMaxwell Student, Cavendish Laboratory^ Fellow of Trinity
College^ Cambridge. Communicated hy Professor J. J. Thomson, F.R,S. 259
VIII. T/ie Crystalline Structure of Metals. (Second Paper.) By J. A. EwiNG,
F.R.S. , Professor of Mechanism and Applied Mechanics in the University of
Cambridge, and Walter Rosenhain, B.A., St. John's College, Cambridge^
\^5\ Exhibition Research Scholar, University of Melbourne .... 279
IX. Lilies of Induction in a Magnetic Field. By Professor H. S. HeleShaw,
LL.D., F.R.S., and Alfred Hay, B.Sc 303
X. On tlie Application of Fourier's Double Integrals to Optical Problems. By
Charles GtODFREY, B.A., Scholar of Trinity, Isaac Neioton Student in the
University of Cambridge. Communicated by Professor J. J. Thomson,
F.R.S. 329
XL An Experimental Investigation into the Flow of Marble. By Frank Dawson
Adams, M.Sc, Ph.D., F.G.S., Logan Professor oj Geology in McGill
University, and John Thomas Nicolson, D.Sc, M.Inst., C.E., Head of the
Engineering Departmenty Manchester Municipal Technical School {formerly
Professor of Mechanical Engi7ieeri7ig in McGill University) .... 363
Index to Volume 403
Erratum 405
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LIST OF ILLUSTRATIONS.
Plates* 1 and 2. — Dr. A. A. Rambaut on the Underground Temperature at Oxford
in the year 1899, as determined by Five Platinumresistance Thermometers.
Plates 3 to 13. — Professor J. A. Ewing and Mr. W. Rosenhain on the Crystalline
Structure of Metals.
Plates 14 to 21. — Professor H. S. HeleShaw and Mr. Alfred Hay on Lines of
Induction in a Magnetic Field.
Plates 22 to 25. — Professor F. D. Adams and Dr. J. T. Nioolson on an Experimental
Investigation into the Flow of Marble.
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[ vii ]
ADVERTISEMENT,
The Committee appointed by the Royal Society to direct the publication of the
Philosophical Transactions take this opportunity to acquaint the public that it fuDy
appears^ as well from the Councilbooks and Journals of the Society as from repeated
declarations which have been made in several former Transactions^ that the printing of
them was always, from time to time, the single act of the respective Secretaries till
the Fortyseventh Volume ; the Society, as a Body, never interesting themselves any
further in their publication than by occasionally recommending the revival of them to
some of their Secretaries, when, from the particular circumstances of their affairs, the
Transactions had happened for any length of time to be intermitted. And this seems
principally to have been done with a view to satisfy the public that their usual
meetings were then continued, for the improvement of knowledge and benefit of
mankind : the great ends of their first institution by the Royal Charters, and which
they have ever since steadily pursued.
But the Society being of late years greatly enlarged, and their communications more
numerous, it was thought advisable that a Committee ot their members should be
appointed to reconsider the papers read before them, and select out of them such as
they should judge most proper for publication in the future Transactions ; which was
accordingly done upon the 26th of March, 1752. And the grounds of their choice are,
and will continue to be, the importance and singularity of the subjects, or the
advantageous manner of treating them ; without pretending to answer for the
certainty of the facts, or propriety of the reasonings contained in the several papers
so published, which must still rest on the credit or judgment of their respective
authors.
It is likewise necessary on this occasion to remark, that it is an established rule of
the Society, to which they will always adhere, never to give their opinion, as a Body,
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[ ^^iii ]
upon any subject, either of Nature or Art, that comes before them. And therefore the
thanks, which are frequently proposed from the Chair, to be given to the authors of
such papers as are read at their accustomed meetings, or to the persons through whose
hands they received them, are to be considered in no other light than as a matter of
civility, in return for the respect shown to the Society by those communications. The
like also is to be said with regard to the several projects, inventions, and curiosities of
various kinds, which are often exhibited to the Society ; the authors whereof, or those
who exhibit them, frequently take the liberty to report, and even to certify in the
public newspapers, that they have met with the highest applause and approbation.
And therefore it is hoped that no regard will hereafter be paid to such reports and
public notices ; which in some instances have been too lightly credited, to the
dishonour of the Society.
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[ ix ]
1900.
List of Institutions entitled to receive the Philosophical Transacteons or
Proceedings of the Royal Society.
Institutions marked a are entitled to receive Philosophical Transactions, Series A, and Proceedings.
„ „ B „ „ „ „ Series B, and Proceedings.
„ AB „ „ „ „ Series A and li, and Procee<ling«.
„ „ /> „ „ Proceedings only.
America (Central).
Mexico.
p. Sociedad Cientifica " Antonio Alzate."
Ainerica(North). (See United States and Canada.)
America (South).
Buenos Ayres.
A6. Mnseo Naclonal.
Caracas.
B. University Library
Cordova.
AB. Academia Nacional de Ciencias.
Demerara.
p, Hoyal Agricultural and Commercial
Society, British Guiana.
LiQ. Plata.
B. Museo de La Plata.
Rio de Janeiro.
p. Observatorio.
Australia.
Adelaide.
p. Royal Society of South Australia.
Brisbane.
^. Royal Society of Queensland.
Melbourne.
p. Observatory.
p. Royal Society of Victoria.
AB. University Library.
Sydney.
Australian Museum.
Geological Survey.
Linnean Society of New South Wales.
Royal Society of New South Wales.
University Library.
P
P
P
AB.
AB*
Austria.
Agram.
p, Jugoslavenska Akademija Znanosti i Um<
jetnosti.
p. Societas HistoricoNaturalis Croatica. I
VOL. CXCV. — A. b
Anstria (continued).
Briinn.
AB. Naturforachender Verein.
Gratz.
AB. Naturwissenschaftlicher Verein fur Steier
mark.
Innsbinick.
AB. Das Ferdinandeum.
p, Naturwissenschaftlich  Medicinischer
Verein.
Prague.
AB. Konigliche Bohmisclie Gesellschaft dcr
Wissenschaften.
Trieste.
B. Museo di Storia Naturale.
p. Societa Adriatica di Scienze Naturali.
Vienna.
Antbropologische Gesellschaft.
Kaiserliche Akademie der Wissenschaften.
K.K. Geographische Gesellschaft.
K.K. Geologische Reichsanstalt.
K.K. Naturhistorisches Hof Museum.
K.K. ZoologischBotanische Gesellschaft.
Oesterreichische Gesellschaft fiir Meteoro
logie.
Von Kuffner'sche Stemwarte.
Belgium.
Brussels.
B. Academic Roy ale de Medecine.
Academic Royale des Sciences.
Mus^ Royal d'Histoire Naturelle de
Belgique.
Observatoire Royal.
Soci6t6 Beige de Geologic, de Paleonto
logie, et d'Hydrologie.
Societe Malacologique de Belgique.
P
aB.
P'
AB.
B.
B.
P'
A.
AB.
P
P'
P'
Ghent.
AB. Univeisite.
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\..
[ X J
Belgium (continued).
Li^ge.
AB. Soci^te des Sciences.
p. Soci^te Geologique de Belgique,
Lonvain.
B. Laboratoire de Microscopic et de Biologie
Cellulaire,
AB. Universite.
Canada.
Predericton, N.B.
p, Universitj of New Brunswick.
Halifax, N.S.
p. Nova Scotian Institute of Science.
Hamilton.
p, Hamilton Association.
Montreal.
AB. McGill University.
p. Natural History Society.
Ottawa.
AB. Geological Survey of Canada.
AB. Royal Society of Canada.
St. John, N.B.
p. Natural History Society.
Toronto.
p. Toronto Astronomical Society.
p, Canadian Institute.
AB. University.
Windsor, N.S.
p. King's College Libi*ary.
Cape of Oood Hope,
A. Observatory.
AB. South African Library.
Ceylon.
Colombo.
B. Museum.
China.
Shanghai.
p. China Branch of tlie Royal Asiatic Society.
Denmark.
Copenhagen.
AB. Kongelige Danske Videnskabernes Selskab.
Egypt.
Alexandria.
AB. Biblioth^que Municipale.
England and Wales.
Aberysiwith.
AB. Univeisity College.
Bangor.
AB. University College of Noith Wales.
Birmingham.
AB. Free Central Library.
AB. Mason College.
p. Philcsophical Society.
England and Wales (continued).
Bolton.
p. Public Library.
Bristol.
p. Merchant Venturers' School.
AB. Univei*8ity College.
Cambridge.
AB. Philosophical Society.
p. Union Society.
Cardiff.
AB. University College.
Cooper's Hill.
AB. Royal Indian P]ngineering College.
Dudley.
p, Dudley and Midland Geological aud
Scientific Society.
Essex.
p. Essex Field Club.
Falmouth.
p. Royal Cornwall Polytechnic Society.
Greenwich.
A. Royal Observatory.
Kew.
B. Royal Gardens.
Leeds.
p. Philosophical Society,
AB. Yorkshire College.
Liverpool.
AB. Free Public Library.
p, Literary and Philosophical Society.
A. Observatory.
AB. University College.
London.
AB. Admiralty.
p. Anthropological Institute.
AB. Board of Trade (Electrical Standards
Laboratory).
AB. British Museum (Nat. Hist.).
AB. Chemical Society.
A. City and Guilds of London Institute.
p. '* Electrician," Editor of the.
B. Entomological Society.
AB. Geological Society.
AB. Geological Survey of Gi'eat Britain.
p. Geologists' Association.
AB. Guildhall Library.
A. Institution of Civil Engineers.
p. Institution of Electrical Engineers.
A. Institution of Mechanical Engineers.
A. Institution of Naval Architects.
p. Iron and Steel Institute.
AB. King's College.
B. Linnean Society.
AB. London Institution.
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[ xi ]
,
f
1
^'
England and Wales (continued).
London (continued).
London Library.
Mathematical Society.
Meteorological Office.
Odontological Society.
Pharmaceutical Society.
Physical Society.
Quckett Microscopical Club.
Royal Agricultural Society.
Royal Astronomical Society.
Royal College of Physicians.
Royal College of Surgeons.
Royal Eiigiueera (for Libraries abroad, six
copies).
Royal Engineers. Head Quarters Library.
Royal Geographical Society.
Royal Horticultural Society.
Royal Institute of British Architects.
Royal Listitution of Great Britain.
Royal Medical and Chirurgical Society.
Royal Meteorological Society.
Royal Microscopical Society.
Royal Statistical Society.
Royal United Service Institution.
Society of Arts.
Society of Biblical Archaeology.
Society of Chemical Industry (London
Section).
Standard Weights and Measures Depaii
ment.
The Queen's Library.
The War Office.
University College.
Victoria Institute.
Zoological Society.
Manchester.
AB. Free Library.
AB. Literary and Philosophical Society.
p. Geological Society.
AB. Owens College.
Netley.
p. Royal Victoria Hospital.
Newcastle.
Free Library.
North of England Institute of Mining and
Mechanical Engineers.
p. Society of Chemical Industry (Newcastle
Section) .
Norwich.
p, Norfolk and Norwich Literary Institution.
Nottingham.
AB. Fi'ee Public Library.
A.
P'
P'
V'
V
p
p
A.
B.
B.
AB.
P'
P
P^
AB.
B.
P'
P'
P
AB.
AB.
P
P'
AB.
AB.
AB.
B.
AB.
P'
England and Wales ^continued).
Oxford.
p, Ashmolean Society.
AB. Radcliffe Library.
A. Radcliffe Observatory.
Penzance.
p. Geological Society of Cornwall.
Plymouth.
B. Marine Biological Aflsociation.
p. Plymouth Institution.
Richmond.
A. " Kew " Observatory.
Salford.
p. Royal Museum and Library.
Stonyhurst.
p. The College.
Swansea.
AB. Royal Institution.
Woolwich.
AB. Royal Artillery Library.
Finland.
Helsingfors.
p. Societas pro Fauna et Flora Fennica.
AB. Soci^te des Sciences.
France.
Bordeaux.
P
V'
P
P
Academic des Sciences.
Faculte des Sciences.
Soci6t^ de M6decine et de Chirurgie.
Soci^t^ des Sciences Physiques et
Naturelles.
Caen.
p. Soci^t^ Linn6enne de Normandie.
Cherbourg.
p. Soci6te des Sciences Naturelles.
Dijon.
p. Academic des Sciences.
Lille.
p. Faculty des Sciences.
Lyons.
AB. Academic des Sciences, Belles  Let tres et Arts.
AB. University.
Marseilles.
AB. Faculte des Sciences.
Montpellier.
AB. Academic des Sciences et Lettres.
B. Faculty de M6decine.
Nantes.
P
Paris.
AB.
P
P'
Soci^te des Sciences Naturelles de TOuest
de la France.
Academic des Sciences de Tlnstitut.
Association Fran9ai8e pour TAvancement
des Sciences.
Bureau des Longitudes.
h 2
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[ xvi ]
Qnited States (continued).
Philadelphia.
AB. Academy of Natural Sciences.
AB. American Philosophical Society.
p. Franklin Institute.
p. Wagner Free Institute of Science.
Rochester (N.Y.).
p. Academy of Science.
St. Louis.
p. Academy* of Science.
Salem (Mass.).
p. American Association for the Adyance
ment of Science.
AB. Essex Institute.
San Francisco.
AB. California Academy of Sciences.
United States (continued).
Washington.
AB. Patent Office.
Smithsonian Institution.
United States Coast Surrey.
United States Commission of Fish and
Fisheries.
United States Geological Survey.
United States Nayal Observatory.
United States Department of Agriculture.
United States Department of Agriculture
(Weather Bureau).
West Point (N.Y.)
AB. United States Military Academy.
AB.
AB.
B.
AB.
AB.
P'
A.
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I xvii ]
Adjudication of the Medals of the Royal Society for the year 1900,
by the President and Council.
The COPLEY MEDAL to Professor Marcellin Berthelot, For. Mem.R.S., for
his brilliant services to Chemical Science.
The RUMFORD MEDAL to Professor Antoine Henri Becquerel, for his
discoveries in Radiation proceeding from Uranium.
A ROYAL MEDAL to Major Percy Alexander MacMahon, F.R.S., for the
number and range of his contributions to Mathematical Science.
A ROYAL MEDAL to Professor Alfred Newton, F.R.S., for his eminent
contributions to the science of Ornithology and the Geographical Distribution of
Animals.
The DAVY MEDAL to Professor Guglielmo Koerner, for his brilliant investi
gations on the Position Theory of the Aromatic Compounds.
The DARWIN MEDAL to Professor Ernst Haeckel, for his longcontinued
and highlyimportant work in Zoology, all of which has been inspired by the spirit
of Darwinism.
The Bakerian Lecture for the year 1900, " On the Specific Heat of Metals
and the Relation of Specific Heat to Atomic Weight," was delivered by Professor W.
A. Tilden, F.R.S., on March 8, 1900.
The Croonian Lecture for the year 1900, " On Immunity with Special Reference
to Cell Life," was delivered by Professor Dr. Paul Ehrlich, on March 22, 1900.
VOL. CXCV. — A. C
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PHILOSOPHICAL TRANSACTIONS.
I. Mathematical Contributions to the Theoiy of Evolution. — VII. On the
Correlation of Characters not Quantitatively Measurable.
By Karl Pearson, F.R.S.
{From tlie Deparimeitt of Applied Mathematics^ University College^ London,)
Eeceived February 7,— Read March 1, 1900.
NOTE.
In August, 1899, I presented a memoir to the Boyal Society on the inheritance of coatcolour in the
horse and of eyecolour in man, which was read November, 1899, and ultimately ordered to be published in
the * Phil. Trans.' Before that memoir was printed, Mr. Yule's valuable memoir on Association was read,
and, further, Mr. Lesue BrahleyMoore showed me that the theory of my memoir as given in § 6 of the
present memoir led to somewhat divergent results according to the methods of proportioning adopted.
We therefore undertook a new investigation of the theory of the whole subject, which is embodied in the
present memoir. The data involved in the paper on coatcolour in horses and eyecolour in man have all
been recalculated, and that paper is nearly ready for presentation.^ But it seemed best to separate the
purely theoretical considerations from their application to special cases of inheritance, and accordingly the
old memoir now reappears in two sections. The theory discussed in this paper was, further, the basis of a
paper on the Law of Eeversion with special reference to the Inheritance of Coatcolour in Basset Hoimds
recently communicated to the Society, and about to appear in the * Proceedings.*!
While I am responsible for the general outlines of the present paper, the rough draft of it was
taken up and carried on in leisure moments by Mr. Leslie BramleyMoore, Mr. L. N. G. Filon, M.A.,
and Miss Alice Lee, D.Sc. Mr. BramleyMoore discovered the wfunctions ; Mr. Filon proved most of
their general properties and the convergency of the series ; I alone am responsible for sections 4, 5, and 6.
Mr. Leslie BramleyMoore sent me, without proof, on the eve of his departure for the Cape, the
general expansion for z on p. 26. I am responsible for the present proof and its applications. To Dr.
Alice Lee we owe most of the illustrations and the table on p. 17. Thus the work is essentially a
joint memoir in which we have equal part, and the use of the first personal pronoun is due to the fact
that the material had to be put together and thrown into form by one of our number, — K P.
Contents.
page
§ 1. On a General Theorem in Normal Correlation for two Variables. Series to Determine the
Correlation 2
§ 2. Other Series for the Determination of the Correlation 7
* Since ordered to be printed in the * Phil. Trans.'
t Bead January 25, 1900. ' Roy. Soc. Proc.,' vol. 66, p. 140.
VOL. CXCV.— A 262. B 16.8.1900.
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2 PROFESSOR K. PEARSON ON iMATHEMATICAL CONTRIBUTIONS
§ 3. Proof of the General Convergency of the Series for the Correlation 10
§ 4. On the Probable Error of the Correlation thus Determined 10
§ 5. To Determine a Physical Meaning for the Series and on Divers Measures of Association ... 14
§6. On the " Excess " and its Relation to Correlation and Relative Variability 18
§ 7. On a Generalisation of the Fundamental Theorem of the present Memoir. Special Formulae
for Triple and Quadruple Correlation 23
§ 8. Illustrations of the Methods of the Memoir 35
Illustration I. Inheritance of Coatcolour in Thoroughbred Horses. Sire and Filly 35
„ II. Chance that an Exceptional Man is bom of an Exceptional Father 37
„ III. Inheritance of Coatcolour in Dogs, Half" Siblings " 38
„ IV. Inheritance of Eyecolour between Maternal Grandmother and Granddaughter . . 39
„ V. Inheritance of Statiu^e between Father and Son for different groupings 40
„ VI. Correlation between Strength to resist Smallpox and Degree of Effective Vaccination 43
„ VII. Effect of Antitoxin on Diphtheria Mortality 44
„ VIII. Chance of Stock above the Average giving Produce above the Average as compared
with the chance of such Produce from Stock below the Average 45
„ IX. Chance of an Exceptional Man being bom of Exceptional Parents 46
§ (1.) On a General Theorem in Normal Correlation.
Let the frequency surface
N
z =
where
27rv/(l  r^)<ri<r^
N = total number of observations,
cTi, 0*2 = standard deviations of organs x and y,
r = correlation of x and y,
be divided into four parts by two planes at right angles to the axes of x and y at
distances h' and k' from the origin. The total volumes or frequencies in these parts
will be represented by a, 6, c, and d in the manner indicated in the accompanymg
plan : —
TdiJb/e of Frequenc/ea
dL
b
<SL'¥b
C
d
c^d
d*C
b^d
N
Then clearly
c? =
27rv/(l
^== f fe »i473(«' + 1"  ^^y) dxdy. .
(1  r»)J» Jt
if
27rv/(i  r») .
h = h'/ai and k = h'/tr^
(i.).
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Further,
and
TO THE THEORY OF EVOLUTION,
N_ r _
*^(fo;
("•).
^ V TT Jo
<^^ti)^^^=vif:e% ■. . . .w.
Thus, when a, 6, c, and dl are known, h and ifc can be found by the ordinary table of
the probability integral, say that of Mr. Sheppard (* Phil. Trans.,' A, vol. 192. p. 167,
Table VI.*). The limits accordingly of the integral for d in (i.) are known.
Now consider the expression
.e*rr^<*' + ^^'^> = U,say, (vi.),
x/1  r«
and let us expand it in powers of r. Then, if the expansion be
/'d»U\
(vii.).
(viii.).
we shall have
Taking logarithmic differentials, we get at once
(1 _ r^)2^ = {xy + r(l  a^  y^) + i^xy  7^}JJ.
Differentiating n times by Leibnitz's theorem, and putting r = 0, we have, after
some reductions
^«+i = ^(2n — 1 — x^ — y*)w^i
 n{n —l){n — 2fun.^
+ ^{un + n{n — l)Un^^} (ix.).
Hence we find
Wo =
= 1
Mi =
xy
«s =
= («»
•1)(3^1)
^3 =
= a;(a:»
3)y(y«3)
W^r
= («*
•6a:^ + 3)(y*.
6y« + 3) J
* See, however, footnote
, p. 5.
B 2
(X.)
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4 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS
Thus the following laws are indicated : —
Un = v„X w„ (xL),
where v, = an;,_i — (n — l)v<,_g (xiL),
■w« = yw^i — (« — l)«'«9 (xiii.).
We shall now show that these laws hold good by induction. . Assume
Thus M,+i = xyu„ + n^u^_i — n(yw,v._i + aw,u>._j).
But by (ix.), substituting for m,_3 from (xi.) and (xiii.),
««+i = ay {v„w„ + w(n — 1)v,_2m;,_2J + n{2n —l—x^ — i/*)i;«_itp,_,
— n{n — l)t\_iw,_i — ycyn{n — l)v„_8w;,_3
+ n(n — 1) (yt',_iu»_2 + xVn^^Wni).
4 n(n — 1) (y v„_iW,_s + a5t;,_gw,_i)
+ a;u>,_i(a;v,_i — n— lv«_8}
= v»+iW,+i, as we have seen above.
Thus, if the theorem holds for u^, it holds for m,+i. Accordingly
where the 7/s and v/b are given by (x.), (xiL), and (xiii.).
It is thus clear that k~ \ I V dx dy consists of a series of which the general
term is
1
n
V«W«r*
1 f*
where V„ = y= e"**'v„da;
'^'''■^\y^«<^
It remains to find these integrals.
The general form of v^ is given by
.. ==af'^%^^af^+ "<  '^%, ^)<"  ^> af*  &c. . . (xv.).
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TO THE THEORY OF EVOLUTION. 5
For this obviously gives (x.). Assume it true for v«_i and v„_2, then
— XT 2U ^ + 222^ * — . . .
= «».
Thus the expression (xv.) is shown to hold by induction, the general terms being
or the general term in u^.
We notice at once that
dx
^r = ^^«i (^^0
Thus, by (xii.)
dx
Vn — XVn^^ —
Multiply by e"*"^ and integrate
Integrating the latter integral by parts, we have
Now y— e"*** can be found from any table of the ordinates of the normal curve,
e.g., Mr. Sheppakd's, * Phil. Trans.,' A, vol. 192, p. 153, Table I.* We shall accord
ingly put
= = C4«'" K=^eH' (xvu.),
and look upon H and K as known quantities.
* For our present purposes the differences of Mr. Sueppa.rd's tables are occasionally too large, but the
following series give very close results : —
Let '^^ = Vl (<.«• c) (6 + d) ^ I V,^ ^ ^y (j^ )^
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6 PEOFESSOR K. PEAESON ON MATHEMATICAL C50NTRIBUTI0NS
Further, let us write (v«i)* » * as v«. j, and similarly (M;«_j)y » * as Ww.,. Thus
V, = H.'U«.„ W, = K.'M>,_i (xviil).
We have then from (i.)
= ill \y''''''<^dy + s(hk^,.,^._,)
(ft + d)(<! + rf) "/r"   \
by (ii.) and (iii.).
Or, remembering that N = a + 6 + c4c?>we can write this
ad — he 2 /r"  \
= r + ^ M + J (A«  1) (i«  1) + gA(A«  3)*(i« 3)
+ r^(A«  15¥ + 45^2  15) (Jfc»  15;fc* + 45*2 _ 15)
5040'
40320'
+ ifti^A(A«  21^* + 105A«  105)ife(P  21^ + 105*^ 105) + , &c.
. . . . (xix,).
Then A = xi + lx.« + ^Xi' + ^Xi' +
and
1 ,75/, 1 2.7 . , 127 . ,
H = ^^"i^ ■*• [2;^^ + [4 >^^ + 76>^^ + ■
1 7 127
* = X2 + j3 X2» + jg X2* + ryX2^ + • • •
1 Ts/, 1 Ys 7 . 127 .
These follow from the considerations that if
Xi = V2«<^.
Xj = V2t*s.
d<h „
t=K.
iH .
#1 ==  *'
whence it is easy to find the successive differentials of h with regard to <f>i and k with regard to <^2> ^nd
then obtain the above residts by Maclaurin's theorem. There is, of course, no difficulty in calculating %
H and K from (xvii.) directly. That method was adopted in the niunerical illustrations. j
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TO THE THEORY OF EVOLUTION. 7
Here the lefthand side is known, and since h and k are known, we can find the
coefficients of any number of powers of r so soon as the first two have been found,
firom (xii.) and (xiii.).
Accordingly the correlation can be found if we have only made a grouping of our
frequencies into the four divisions, a, 6, c, and d.
If h and k be zero, we have fi:om (xvii.) and (iv.)
H = K= ^
i/27r
The righthand side of (xix.) is now ^ . c   it? ^ ^
r+f3r» +
5o <x ■ (L a**^ t  K,.
or equal to sin"^ r.
_. . ^ (ad — 6c)
Hence r — sm 2ir ^^ — rr^ —
= cos IT —2 (XX.),
which agrees with a result of Mr. Sheppard's, * Phil. Trans.,' A, voL 192, p. 141. We
have accordingly reached a generalised form of his result for any classindex whatever.
Clearly, also, r being known, we can at once calculate the fi:equency of pairs of organs
with deviations as great as or greater than h and k.
§ (2.) Other Series for the Determination of r.
For many purposes the series (xix.) is sufficiently convergent to give r for given
h and k with but few approximations, but we will now turn to other developments.
We have by (vii.)
Put X = hf y = k, and write for brevity
ad ^ be
c =
N«HK
It follows at once from (xix.) that
(xxL).
dr
^^0
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8 PKOFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS
— qW gi(it tan «  A«ec «)« ^^
if r = sin 6.
Now either of the quantities under the sign of integration in (xxii.) can be expanded
in powers of 6 by Maclaurin's theorem. Thus let
= Xo + (g)/ + (^)+...+(g)i + ...
Then
and it remains to find ( tx I •
Now log X = — i (^ tan ^ — A sec dy.
Hence
cos* ^ ^ =  X [(A' + ^•*) sin ^ _ M (A  ^ cos 2^)].
Differentiating n — 1 times by Leibnitz's theorem, and putting ^ = 0,
Clearly Xo = ^"**'> then we rapidly find
Or, finally
c = ^ + 1M^2  (A^ + P  A^P) ^ + ;i&{;i.2ifc»  3(A^ + p) + 5} ^ + . . . (xxiv.),
where more terms if required can be found by (xxiii.). If 6 be fairly small, 6^ will be
negligible. Or if A and k be small, the lowest terra in the next factor will be h^ f /:^,
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TO THE THEORY OF EVOLUTION. 9
and this into ^/5 is generally quite insensible/ Very often two or three terms on the
righthand side of (xxiv.) give quite close enough values of ^, and accordingly of
r = sin^. (xxiv.) is clearly somewhat more convergent than (xix.) if ^ and h are, as
usually happens, less than unity.
Returning now to (xix.), let us write it
e=f{r,h,k).
This is the equation that must be solved for r. Suppose r^ a root of this when we
retain only few terms on the right, say a root of the quadratic
€ = r + p.*r*.
Then if r = Tq + /»>
« =/(ro, h, k) + pf{r^, h, k) + \ipY"{r„ h, k) + &c.
Hence p = ^ ./ vH to a third approximation
V^r,
1 ir:^^;^;^^;^^ nearly (XXV.),
which gives us a value of p which, substituted in p^ in the above equation, introduces
only terms of the 6th order in r,^.
Another integral expression for c of Equation (xxi.) may here be noticed :
)v/l
PutA=^(^ + y),A: = ^(^.y)
Hence
Jo\/l — ^
Jo Vi — T*
1 — r
Let tan 2^ = , or, r = co82 if>.
Therefore
f46<»
Jl 1 +V^
where t; = cot and is > 1.
It seems possible that interesting developments for € might be deduced from this
integral expression.
VOL. cxcv.— A. o
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10 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS
§ (3.) To show that the SeiHesfor r is Convergent i/ r < 1, whatever be the Values
of h and k
Write the series in the form of p. 6, i.e. : —
Now
W,+i = Aw. — WW,_i J •' ^ '
From these we deduce
V»+l = {^' — (2W — 1)} Vni — (n — 1) (w — 2) v,_3
^«+i = {** — (2w — 1)} w,_i — (n — 1) (n — 2) w,_3
Now let «» = iLi»**{»}*, «« =~M'»i»'*" {!«}*.
Then we find
^'•+* "" v/(n+l)(7. + 2) ^"^ V 71(71+1) (71 +2; ^'^^'^ '
^■^^  ^(n + l)(7i + 2) ^^ V n(7i + l)(ri + 2) ^'^"^ '
Thus, when 7^ is large, we find the ratio of successive terms Sn^Js^ or t^+cj^n is given
by /o, where
/) = — 2r — r^//o or, /o = — r.
The ultimate ratio of s^+2 '*»+2 *^ ^» '« ^s accordingly given by r*, but this is the
ratio of alternate terms of the original series. The original series thus breaks up
into two series, one of odd and one of even powers of r. Both these series are
absolutely convergent whatever h and k be, having an ultimate convergence ratio of r ^
§ (4.) To find the Probable Error of the Coirelation Coefficient as Determined by the
Method of this Memoir.
Given a division of the total frequency N into a, 6, c, d groups, where
a + & + c + d = N, then the probable error of any one of them, say a, is '67449 cTa,
where*
<r«= y/^^ (xxvi.).
Let 6 + c? = nj, c + d = 7^2, then
* The standard deviation of an event which happens np times and fails nq times in n trials is well
known to be Jnpq. The probable errors here dealt with are throughout, of course, those arising from
different samples of the same general population.
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TO THE THEORY OP EVOLUTION. 11
. = a/^^^ <r. = ^/^^ . . . (xxvil).
To obtain Vcd we have, if Br/ denotes an error in any quantity tj,
Sc + 8d = Sng,
.. aJ^ + <r/ + 2a'ca'drcd = <r«.® (xxviii.),
by squaring, summing for all possible variations in c and rf, and dividing by the total
number of variations.
Hence, substituting the values of the standard deviations as foxmd above, we
deduce
ocOdTed = — ccZ/N (xxix.).
In a similar manner
hn^U=hhU + (U)\
O'dO'n^rdn^ = OhOdTbd + Ct/
orfo^^r^^ = d (a + c)/N (xxx.).
and <rrfO„,ri/,^= ci (a + &)/N (xxxi.),
N f*
Now ^1 ~ "TT"] ^"^^^
Thus (r^=NH(rA (xxxiL),
and similarly Cn^ = NK o^ . . (xxxiii.).
Hence the probable error of h
=^ ^25^13 (^,w.),
andofi =^^<I±^±I). ..... .(X.ZV.).
They can be found at once, therefore, when H and K have been found from an
ordinate table of the exponential durve, and a, 6, c, d are given. We have thus the
probable error of the means as found from any double grouping of observations.
Next, noting that
8«iK = N*HK8A8Jfc,
we have (r„, <^njrn,H^ — N^HK Ck okr^,
or Tn^n, = Tkh.
c 2
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12 PEOFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS
But 8ni 8% = (S6 + Sd) (8c + 8d),
= j^ . (xxxyi.),
therefore
<Tk(Tkrhk— ygj^ (xxxvil).
ad — he , ... .
^^~ v/(T+rf)(aHc)(c + ei)(a + 6) • • • • • (xxxvui.).
This is an important result ; it expresses the correlation between errors in the
position of the means of the two characters under consideration. But if the prob
abilities were independent there could be no such correlation. Thus r;^ might be
taken as a measure of divergence from independent variation. We shall return to
this point later.
Since S^i = — HNSA, we have Sn^Sci = — HNSrfS^, whence we easily deduce
rdn,— rah (xxxix.).
Similarly ^if«,= '^'^dk • . (xl.).
Now d is a function of r, h^ and k. Hence if d = f{r^ A, k)^
8rf=^8r + f 8^ + 8ik
dr dh ok
= 70^^ + 71^^ + 72^* (xli.)
Whence transposing, squaring, summing, and dividing by the total number of
observations, we find
yo^or^ = cr/ + y^o^ + y^oj?' — ^y^adOkrdh — ^y^odOkTdk
+ "^yiy^ohokThk
Substituting the values of the standard deviations and correlations as found above,
we have
V» = j^ { c/(a + 6 + c) + (^^(a + h) {d + c) + (^)'(a + c) (d + b)
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TO THE THEORY OF EVOLUTION. 13
It remains now to determine yo, yi, and y^.
By Equation (i.)
' _df N
= iiJ;l>"^^ (!)>
where A= ^7r« 
Thus
= '/'2i (xlv.).
Sinailarly •/^/(NK) = ^^ — ^ . . . ^ . . . . (xlvi.).
H^^^ ^^ = ;7feD"*'^^' ^^=:^D"''^ • • • (^^^^^
where A= yfJ^ > ^^ = TT^ (xlviii.),
and thus t/r^ and i/r^ can be found at once from the tables when fi^ and ^82 are found
from the known values of r, h^ k.
Lastly, we have from Equation (xxi.)
d (d + h)(d + e) . 1 f' TJ
Thus* y^ = d//dr = ~l^U,
ro/N = xo.
where Xo = ^ ^i^ e.ir^ <»» + ***") (xlix.)
a value which can again be foiind as soon as r, ^, A; are known, y^ s ^^ is clearly
the ordinate of the frequency surface corresponding to x =^ h,y ^= L
Substituting in Equation (xliiL) we have, after some reductions,
* By Equations (ii.) and (iii), d + b and d + e are independent of r.
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14
PEOFESSOR K. PEAESON ON MATHEMATICAL CONTRIBUTIONS
Probable error of r s= •67449<rr
•67449 r (a + d)(e+ b) ^ (a + c)(d + b) ^ (<^ + b)(d+e)
y/K^ \ 4N« "^ '''» N» "^ '''1 N»
ad^hc 'ah — ed , ac — 6d 1 i
N»
(1).
where Xo> ^i> ^^^ ^a ^^^ readily found from Equations (xlix.), (xlvii.), and (xlviii.).
' Thus the probable error of r can be fairly readily found. It must be noted in using this
formula, that a is the quadrant in which the mean falls, so that h and k are both
positive (see fig., p. 2). In other words, we have supposed a + c > 6 + r? and
a + 6 > c + d. Our lettering must always be arranged so as to suit this result
j before we apply the above formula.
§ (5.) To Find a Physical Meaning for the Series in ?% or for the € of Equation (xxL).
I h
Return to the original distribution ^ , of p. 2. If the probabilities of the two
characters or organs were quite independent, we should expect the distribution
N
a\l a •{ c
N N
N
c •{ d ai c
N
g f ft b '\' d
N N
N
ch db + d
N N
N N
Now rearranging our actual data we may put it thus
a i b b + d
a I b
N N
N
N
e + d a + c ad — be
N (N
N
ad— be
n:
N N
ad be
T^Tc + dft + d , orf — 6c
N
N
N
Accordingly correlation denotes that — jx— has been transferred from each of the
second and fourth compartments, and the same amount added to each of the first and
third compartments. If t^ = {ad — 6c)/N^ then ri is the transfer per unit of the total
frequency. The magnitude of this transfer is clearly a measure of the divergence of
the statistics from independent variation. It is physically quite as significant as the
correlation coeflEicient itself, and of course much easier to determine. It must vanish
with the correlation coefficient. We see from (xxi.) that
ri = €X HK,
or we have an interpretation for the series in r of (xix.).
Now, obviously any function of 17, just like ri itself, would serve as a measure 01
the divergence from perfectly independent variation. It is convenient to choose a
function which shall lie arithmetically between and !•
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TO THE THEORY OF EVOLUTION. 15
Now consider what happens in the case of perfect correlation, i.e., all the observa
tions fall into a straight line. Hence if ad > 6c, either 6 or c is zero, for a straight
line cannot cut all four compartments, and a and d are obviously positive. Thus c
and 6 can only be zero if i/ = (c + d){a + c)/N^ or (a + h){b + d)fN\ In order
that 6 should be zero, it is needful that h and A, as given by (iv.) and (v.), should be
positive ora+c>6+d, a + 6>c + c?, and the mean fall under the 45° line
through the vertical and horizontal lines dividing the table into four compartments,
i.e.y h > k. These conditions would be satisfied if ad >hc and a > d^ c > b. Now
suppose our fourcompartment table arranged so that
ad>hCy a>d, c>by
and consider the function
Q' = ^^^f(a + 6)(5 + cO/N' (^•)'
or
^^ . TT ad — be ,,,. V
Q' = "^"2 (a.f&)(5 + rf) <^>
This ftmction vanishes if 17 = 0, and it further = unity if 6 = 0. Thus it agrees
at the limits and 1 with the value of the correlation coefficient. Again, when h
and k are both zero, a = d, 6 = c, and Q^ = sin n ^ . ^ , is thus r by (xx.). Hence
we have found a ftinction which vanishes with r and equals unity with r, while it is
also equal to r if the divisions of the table be taken through the medians.
Now, I take it that these are very good conditions to make for any function ot"
a, 6, c, d which is to vanish with the " transfer," and to serve as a measure of the
degree of dependent variability, or what Mr. Yule has termed the degree of
** association." Mr. Yule has selected for his coefficient of association the expression
QCUL ""^ OC /« . • • *
This vanishes with the transfer, equals unity if 6 or c be zero, and minus unity if a
or (i be zero. The latter is, of course, unnecessary if we agree to arrange a, 6, c, d
so that ad is always greater than be. Now it is clear that Q2 possesses a great
advantage over Q^ in rapidity of calculation, but the coefficient of correlation is also
a coefficient which measures the association, and it is a great advantage to select one
which agrees to the closest extent with the correlation, for then it enables us to
determine other important features of the system.
If we do not make all the above conditions, we easily obtain a number of coeffi
cients which woyld vanish with the transfer. Thus for example the correlation Vkjt of
Equation (xxxviii.) is such an expression.* It has the advantage of a synmietrical
. form, and has a concise physical meaning. It does not, however, become .imity when
* In fact (xxxvii) gives us c » ajiO'tirkk'
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16 PROFESSOR K, PEARSON ON MATHEMATICAL CONTRIBUTIONS
either, but not both, 6 and c vanish, nor does it, unless we multiply it by 7r/2 and
take its sine, equal the coefficient of correlation when a ^=^ d and b = c.
Again, we might deduce a fairly simple approximation to the coefficient of correla
tion from the Equation (xxiv.) for ^, using only its first few terms. Thus we find
f>*. dd — he /!• \
8m 27r ^,^^ _ ^^^, ^ ^^.^^ ^ ^^^^^^^ ^ ^^ (liv.),
where Xi = V 2 N '
X^ "" V 2 N '
as an expression which vanishes with the transfer, and will be fairly close to the
coefficient of correlation. It is not, however, exactly unity when either 6 or c is
zero. But without entering into a discussion of such expressions, we can write
several down which fully satisfy the three conditions : —
(i.) Vanishing with the transfer.
(ii.) Being equal to unity if 6 or c = 0.
(iii.) Being equal to the correlation for median divisions.
Such are, for example : —
Q« = smjf:j : 26c ^ ,od>}>c . . . (Ivi.),
{ad — be) (b + c)
where i^ =
2 ^1
Aahcd W
(ad'bcy(aid)(b + c)
Only by actual examination of the numerical results has it seemed possible to pick
out the most efficient of these coefficients. Q^ was found of little service. The
following table gives the values of Qg, Q3, Q4, and Q5 in the case of fifteen series
selected to cover a fairly wide range of values : —
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17
No.
r.
h.
k.
Q2.
Qs.
04.
0*.
1
•5939 ± 0247
 •0873
 ^4163
•7067
•6054
•6168
•6100
2
•5557 ± 0261
 ^4189
 4163
•6688
•5657
•5405
•5570
3
•5529 ± ^0247
 •0873
 ^0012
•6828
•5809
•5699
•5813
. 4
•5264 ± 0264
+ ^2743
+ •3537
•6345
•5331
•5200
•5283
5
•5213 ± ^0294
+ ^6413
+ ^6966
•6530
•5511
•4878 •
•5160
6
•5524 ± 0307
+ r0234
+ 3537
•7130
•6118
•6169
•6138,
7
•5422 ± 0288
+ ^6463
+ ^5828
•6693
•5673
•5136
•5452
8
•2222 ± •0162
+ ^3190
+ ^3190
•2840
•2268
•2164
•2251
9
•3180 ± ^0361
+ ^1381
+ ^0696
•3959
•3185
•3176
•3183
10
•5954 ± ^0272
+ 1^5114
+ •7414
•7860
•7100
•6099
•6803 
11
•4708 ± 0292
+ ^0865
 0054
•5692
•4712
•4720
•4715
12
•2335 ± •0335
+ ^0405
+ ^0054
•2996
•2385
•2385
•2385
13
•2451 ± ^0205
+ 2707
+ ^0873
•3103
•2473
•2456
•2470
14
•1002 ± ^0394
+ 4557
+ ^1758
•1311
•1032
•0993
•1029
15
•6928 ± ^0164
+ 5814
+ 5814
•8032
•7108
•6699
•6897
Now an examination of this table shows that notwithstanding the extreme ele
gance and simplicity of Mr. Yule's coefficient of association Q^, the coefficients Q3,
Qt, and Q5, which satisfy also his requirements, are much nearer to the values
assumed by the correlation. I take this to be such great gain that it more than
counterbalances the somewhat greater labour of calculation. If we except cases (6)
and (10), in which h or k take a large value exceeding unity, we find that Q3, Q^, and
Qg in the fifteen cases hardly differ by as much as the probable error from the value
of the correlation. If we take the mean percentage error of the difference between
the correlation and these coefficients, we find
Mean difference of Q^ = 24*38 per cent.
» >> ^3 ^^ 3'9o ,,
„ Q^= 294
,, Q5= 272
Thus although there is not much to choose between Q^ and Q5, we can take Q5 as
a good measure of the degree of independent variation.
The reader may ask : Why is it needful to seek for such a measure ? Why cannot
we always use the correlation as determined by the method of this paper ? The
answer is twofold. We want first to save the labour of calculating r for cases where
the data are comparatively poor, and so reaching a fairly approximate result rapidly.
But laboursaving is never a wholly satisfactory excuse for adopting an inferior
method. The second and chief reason for seeking such a coefficient as Q lies in the
fact that all our reasoning in this paper is based upon the normality of the frequency.
We require to free ourselves from this assumption if possible, for the difficulty, as
is exemplified in Illustration V. below, is to find material which actually obeys
within the probable errors any such law. Now, by considering the coefficient of
regression, raJc^ = 8{xy)/Q^a'ia'^), as the slope of the line which best fits the series
VOL. CXCV, — ^A. D
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1 8 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS
of points determined as the means of arrays of x for given values of y, we have once
and for all freed ourselves from the difficulties attendant upon assuming normal
frequency. We become indifferent to the deviations from that law, merely observing
how closely or not our means of arrays fall on a line. When we are not given arrays
but gross grouping under certain divisions, we have seen that the " transfer " is also
a physical quantity of a significance independent of normality. We want accordingly
to take a function which vanishes with the transfer, and does not diverge widely
from the correlation in cases that we can test. Here the correlation is not taken as
something peculiar to normal distributions, but something significa^t for all distribu
tions whatever. Such a function of a suitable kind appears to be given by Qg.
§ 6. On the " Excess " and its Relation to Correlation and Relative Variability.
There is another method of dealing with the correlation of characters for which
we cannot directly discover a quantitative scale which deserves consideration. It
is capable of fairly wide application, but, unlike the methods previously discussed, it
requires the data to be collected in a special manner. It has the advantage of not
applying only to the normal surface of frequency, but to any surface which can be
converted into a surface of revolution by a slide and two stretches.
It is well known that not only the normal curve but the normal surface has a
type form from which all others can be deduced by stretching or stretching and
sliding. Thus in 1895 the Cambridge Instrument Company made for the instrument
room at University College, London, a " biprojector," an instrument for giving
arbitrary stretches in two directions at right angles to any curve. In this manner
by the use of typetemplates we were able to draw a variety of curves with arbi
trary parameters, c.gr., all ellipses from one circle, parabolas from one parabola,
normal curves from one normal curve template. Somewhat later Mr. G. U. Yule
commenced a model of a normal frequency surface on the Brill system of inter
laced curves. This, by the variable amount of slide given to its two rectangular
systems of normal curves, illustrated the changes from zero to perfect correlation.
This model was exhibited at a College soirSe in June, 1897. Greometrically this
property has been taken by Mr. W. F. Sheppard as the basis of his valuable paper
on correlation in the *Phil. Trans.,' A, vol. 192, pp. 101167. It is a slight addition
to, and modification of, his results that I propose to consider in this section.
The equation to the normal frequency surface is, as we have seen in § 1,
_ N r /^ 2rxy , y^\ 1 ]
Now write a;/(<riN/l— r^) = x\ y/c^ = y\ This is merely giving the surface two
uniform stretches (or squeezes) parallel to the coordinate axes. We have for the
frequency of pairs lying between x^ x + Sx, and y, 8 + 8//,
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TO THE THEORY OF EVOLUTION. 19
Now give the area a uniform slide parallel to the axis of x defined by r/y/l— r*
at unit distance from that axis. This will not change the basal unit of area
8a = 8x'By\ and analytically we may write
Whence we find ^ ^ N ^ , ti^v
zoxoy = 2^ 8a expt. (— ^^K*).
This is the mechanical changing of the YuleBrill model analytically represented.
The surface is now one of revolution, and the proof would have been precisely the
same if we had written in the above results any function/, instead of the expo
nential.* It is easy to see that any volume cut off by two planes through the ^is of
the surface is to the whole volume as the angle between the two planes is to four right
angles. Further the corresponding volumes of this surface and the original surface
are to each other as unity to the product of the two stretches. Lastly, any plane
through the zaxis of the original solid remains a plane through the 2:axis after the
two stretches and the slide. These points have all been dealt with by Mr. Sheppard
(p. 101 et seq.y loc. cit.). I will here adopt his notation r = cosD, and term with him
D the divergence. Thus cot D is (in the language of the theory of strain) the slide,
and D is the angle between the strained positions of the original x and y directions.
Now consider any plane which makes an angle x with the plane of xz before strain.
Then, since the contour lines of the correlation surface are ellipses, the volumes of
the surface upon the like shaded opposite angles of the plan diagram below will be
equal ; and if they be n^ and n^, then n^ + n^ = ^N. If n{ and n^' be the volimies
after strain, then by what precedes we shall have
and (rig  ni)/(ni + n^) = «  <)/« + <)•
a?
8
* The generalisation is not so great as might at first appear, for I have convinced myself that this
property of conversion into a surface of revolution by stretches and slides does not hold for actual cases
of markedly skew correlation.
d2
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20 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS
Now Ui and n^' will be as the angles between the strained positions of the planes
bounding n^ and n2* ^^ Aoes not change its direction. Oy is turned through an
angle 7r/2 — D clockwise, and x becomes x\ say. Hence
<:<::x"+I>:l + x"I+D.
or (V  n^)/(n^ + n/) =  (x" + D)  1.
Let us write E^ = 2(% — n^ and term it the excess for the ycharacter for the
line AB. Then we easily find :
/E, TT , 7r\ . / // . rw\ cot y' + cot D ,1 ... .
It remains to determine tan ^ ' and substitute. The stretches alter tan x into
tan x', such that
tan X = ~^ tan x
Further, by the slide
cot y" = cot y' — cot D = — ,^  ^ cot Y — cot D.
Hence we have by (Iviii.) above
 * (s !) = i.;/Tr? t x/(;;7fci oot X oot D  cof D  l) ,
tan(jUootDa55^ (li^).
\N 2/ (Tg smD ^ ^
or.
Now the excess E^ is the difference of the frequencies in the sum of the strips of
the volume made by planes parallel to the plane yz on the two sides of the plane AB2;
(defined by x)> taken without regard to sign. For on one side of the mean yy this is
ri2 — n^y and on the other — (^i— ^2) Hence we have this definition of E^, the
column excess for any line through the mean of a correlation table : Add up the
frequencies above and below the line in each column and take their differences ivithout
regard to sign^ and their sum is the column excess.
If we are dealing with an actual correlation table and not with a method of
collecting statistics, then care must be taken to properly proportion the frequencies
in the colunm in which the mean occurs, and also in the groups which are crossed by
the line. It is the difficulty of doing this satisfactorily, especially if the grouping, as
in eye and coat colour, is large and somewhat rough, that hinders the effective use of
the method, if the statistics have not been collected ad hoc.
Now let E2 be the row excess for the line AB, defined in like manner, then we have
in the same way
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TO THE THEORY OP EVOLUTION. 21
tan(^j)=:cotD^»^ (Ux>).
\N 2/ aj^ smD ^ ^
Now eliminate o^/cti between (lix.) and (lix.^") ; then
(tan (I I) + oot D) (tan (I f ) + cot D) = „^ .
Whence we deduce
and, therefore,
cot D = cot ^^ ~,
N 2
DEj + Eg TT ' n \
= COS ^ 2 V^)*
Substituting for D in (lix.) we find further
^; = cotxcos(yco8(^) (IxL).
Thus Equations (Ix.) and (Ixi.) give the coefficient of correlation and the relative
variability of the two characters. The latter is, I believe, quite new, the former novel
in form.
If we call nil ^^® frequency in the angle x {^Ox of the figure above), then it is easy
to see that E^ = 2(^2 — n^) = N — 471^, and similarly E^ = N — 4m. Thus
(El + E2)/N = 2(N — 2(ni + mi))/N. But n^ + m^ is the frequency in the first
quadrant. This Mr. Sheppard terms P, while that in the second he terms R. We
have thus (E^ + E2)/N = 2ll/(Il + P), or
p
^ = cos^^7r (Ixii.),
ie.y Mr. Sheppard's fimdamental result* (* Phil. Trans.,' A, vol. 192, p. 141).
We can, of course, get Mr. Sheppard's result directly if we put x = 0, when we
have at once E^ = 2(R — P), Eg = N :;= 2(R + P), and the result follows.
Equation (Ixi.) may also be written in the form
^ = cotxsin(^'2^)/8in(^2^) (Ixiii.).
If we put x^ ^y t^^^ ^1 becomes zero, and the righthand side of (Ixiii.) is
indeterminate. If we proceed, however, to the limit by evaluating the frequency in
an indefinitely thin wedge of angle Xj we reach merely the identity cja^ = oi/o^.
Hence there is no result corresponding to (Ixi.) to be obtained by taking
Mr. Sheppard's case of x = 0.
The following are the values of the probable errors of the quantities involved : —
* In the actual classification of data (Ix.) and (Ixii.) suggest quite different processes. We can apply (Ix.)
wbere (Ixii.) is difficult or impossible, e.g., correlation in shading of birds' eggs from the same clutch.
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22 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS ^'^'^^ r
Probable error of E^ = 67449 x/N (1  Ei*/N») .... {\xvt.)\
E^ = 67449 n/N(1  E^/N*) .... (Ixv.). ^
Correlation between errors in E. and E, =  a/ J! " "^^(S! i! T & Si • • (l^vL).
^ * V (1 + ii/N) (I + Ej/N) ^ '
, Tj , , , . 67449 sin D*/D(^:^D) • /, •• x
Probable error m r = ^ r .... (Irvii.),
where D = ' * ^ {cf. Sheppard, loc. cit., p. 148).
Probable error in ratio Ci/oj =
•67449 o^TTf/, E,«\. oi%A L fn E,2\. 2./E,^\
The application of the method here discussed to statistics without quantitative
scale can now be indicated. If the characters we are dealing with have the same
scale, although it be unknown, then, if the quantitative order be maintained, i.e.,
individuals arranged in order of lightness or darkness of coat or eyecolour, the
diagonal line on the table at 45° will remain unchanged, however we may suppose
parts of the scale to be distorted, for the distortion will be the same at corresponding
points of both axes. Further, if we suppose the mean of the two characters to be the
same, this 45° line will pass through that mean, and will serve for the line AB of the
above investigation. In this case we must take tan ^ = 1, and consequently (Ixi.)
becomes
iTj(r.2 = cos y f )/cos (^fj (Ixix.).
We can even, when the mean is a considerable way off the 45° line, get, in some
cases, good results. Thus, the correlation in stature of husband and wife worked out
by the ordinary product moment process is '2872. But in this case E^ = 382*062
Ej = 806'425, and this gives the correlation '2994. On the other hand, the actual
ratio of variabilities is 1*12, while (Ixix.) makes it 276 ! This arises from the fact
that the errors in E^ and E2, due to the mean being off the 45° line, tend to cancel in
El + E^, but tend in directly opposite directions in the ratio of the cosines. Similarly
the correlation between father and son works out *5666, which may be compared with
the values given in Illustration V. below, ranging from '5198 to '5939. Again,
correlation in eyecolour between husband and wife came out by the excess process
•0986, and by the process given earlier in the present Memoir '1002. But all these are
favourable examples, and many others gave much worse results. We ought really only
to apply it to find cja^ when the means are on the 45° line, as in the correlation of the
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23
same character in brethren, and even in this case the statistics ought to be collected
ad hoc, i.e., we ought to make a very full quantitative order, and then notice for each
individual case the number above and below the type. For example, suppose we had
a diagram of some twentyfive to thirty eye tints in order {e.g., like Bertrand's),
then we take any individual, note his tint, and observe how many relatives of a
particular class — ^brethren or cousins, say — have lighter and how many darker
eyes ; the difference of the two would be the excess for this individual. The same
plan would be possible with horses' coatcolour and other characters. After trying the
plan of the excesses on the data at my disposal for horses' coatcolour and human eye
colour (which were not collected ad hoc), I abandoned it for the earlier method of
this Memoir ; for, the classification being in large groups, the proportioning of the
excess (as well as the differences in the means) introduced too great errors for such
investigations.
§ 7. On a Generalisation of the Fundamental Theorenx of the Present Memoir.
If we measure deviations in units of standard deviations, we may take for the
equation to the correlation surface for n variables
N
z =
(27r)VE
e^W^^)H^)} (Ixx.),
where
K =
1
Til
1
'23.
^n1, 1
^,2
'»i, »
rn
n— 1, n
and 'Rpg is the minor obtained by striking out the ^th row and qth column, rpg is, of
course, the correlation between the pth and ^th variables, and equals r^. S^ denotes
a summation for s from 1 to n, and Sg a summation of every possible pair out of the
n quantities 1 to n.
Now take the logarithmic differential of z with regard to r^ We find
For
dB,/dr^ = 2Kp^
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24 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS
and, generally, whether 5 is or is not = s, or these are or are not = p and g, we
have
dir[R)= n* O^xi.).
This follows thus :
dr„ \ K / R drp, R2 dvp^ " R 1^
R2 '
or we have to show that
djxggt £t]\„i \xpq — Kj»t R^#/ — Rpi* Ry*
dvpq "" R
R«l^lVg — RyjRyjf , R<^Ryy — fiw/ Ryj
~ R "*" R
where jb^R«/ is the minor corresponding to the term r^q in R,^, and q^s$> the minor
corresponding to the term r^p.* But this last result is obvious because R^^ only con
tains Vpq in two places, i.e.y as Vp^ and r^p.
Putting s =s\ we have the other identity required above, i.e.^
drpqXHJ R9
Returning now to the value for   — on the previous page, we see that the two
z dvpq
sum terms may be expressed as a product, or we may put
4=^+«.(l)xs.(l)
Now write
(2Tr)'VK'
z = .„ ... /:rt e*.
Tx 1 dz d^(h . d(b d(b
Hence  — = — — ^ + ^ :^
z dVpg dxp dj:^ dxp dxg
Now differentiate log z with regard to x^. Then
dz d<l>
dosp dXp
* See also Soott, * Theory of Detenninante,' p. 59,
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TO THE THEORY OF EVOLUTION. 25
Thus finally
d,uyixj
= —
dz d4>
djCg do:p
1
z
(Pz
dz
^4> ,
dX^g
dh
d^d^
dxp dxg
dr„
dayfajj •
(Ixxii.).
In other words, the operator djdr^ acting on z can always be replaced by the
operator d^jdxpdxq.
Let d/dppq denote the effect of applying the operator d/dvp^ to «, and putting Vpg
zero after all differentiations have been performed, then the effect of this operator will
be the same as if we used d^jdxpdxq on 2;, putting Vpq zero before differentiation.
Generally, let F be any series of operations like d/dvp^^ then we see that
\drpq' dryq, '
ipq Uipig, dVp..^,,
\ dXj/d.Cq dxpdxq, ' d^Vp,dXg„ ' / (27r)**
(x.»)
Now let F be the function which gives the operation of expanding z by Maclaurin's
theorem in powers of the correlation coefficients, i.e.,
F = e««^'5^)>
then
z = e ^(' .7^^ z = — M' 1^) e  »«'<'^ .
This is the generalised form of result (xiv.) reached above.
Now let Zr. — ■ ^ • ^iSi<''">
nov. iGT, Zq — ^2^^^^ € ,
then z^^ is the ordinate of a frequency surface of the nth order, in which the distribution
of the n variables is absolutely independent. We have accordingly the extremely
interesting geometrical interpretation that the operator
applied to a surface of frequency for n independent variables converts it into a surface
of frequency for n dependent variables, the correlation between the sth and sth
variables being r„,*
^ I should like to suggest to the pure mathematician the interest which a study of such operators would
have, and in particular of the generalised form of projection in hyperspace indicated by them.
VOL. CXCV. — A. E
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26 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS
Expanding, we have
+ • • • +{Sa(»'«'d^,)}"^o + . •. . ■ . . (Ixxiii.).
Our next stage is to evaluate the operation
Hj'*''d^ Co
llet us put
,Vi — x„ ,«8 = a?/ — 1, ,V3 = a;,(a;/ — 3),
and ,Vp = the pth function of «, as defined by (xv.).
Let €, be a symbol such that c/ represents ,Vp. Then we shall show that
^^"•^.j =^o = «o{s,(r,^^^)}"' (Ixxiv.).
We shall prove this by induction.
By (xii.)
,v^+i = X, ,Vm — m ,t;«.i,
or c/"*"^ = X, c/ — m c/""\
and by (xvi.)
^ = m,i;«_,. or — = m e, ^
Now, let X (€*) be any function of c.
if we suppose it can be expanded in powers of c,
Then
d
= S(A,<?e,»i)
= S(A,(x,€,»e/+i))
= a!^(A, €,1)  €. S(A, €/)
= («*«') X(«') (Ixxv.).
Similarly ;^;;j^ x(«* .«'') = (^'«')(^'—«'') x(«« «*') • • • • (Ixxvi.).
Now suppose that
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TO THE THEORY OF EVOLUTION. 27
then
{^«(^d^)}'""^o = S,(^^>oU,
where U stands for. {S2(r,^€,€^)}*.
Hence, remembering that dzjdx^ = — z^t>y
= «oS»(»'«.<»e^)U'
= 2o{S2(r^€^^)}+',
which had to be proved.
But it is easy to show by simple differentiation that
^ Zo {^i{r^^,)Y (Ixxvii.).
Hence the theorem is generally true.
Thus we conclude that
+ • • • + 4 {S2(^'^'*.)}'"+ . • • •] O^^viii.).
It is quite straightforward, if laborious, to write down the expansion for any number
of variables.
Now let Q be the total frequency of complices of variables with x^ lying between
h^ and 00 , X2 between h^ and oo , . . . aj, between h, and oo , . . . 5c„ between h^ and oo ;
and let Qq be the frequency of such complices if there were no correlations.
Then
I ... I ... 2; dx^ dx^ . . . dxt
dx^
fCD •flO ttOO ftO
I ... I ... 1 ZQdx^dxci . . . dx, . . . dXf^
hi JA, Ja. Ja.
Now let
£ 2
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28 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS
where
We have Qo = 'iHfiA . . A • • i8« H^ Hg . . H, . . H,.
But by (xviii.)
where
and as above,
Thus
,W/,_l = [,Uj,_l]^ = ». (Ixxx.),
^, = ^'ei^dx./ei'"\ (Ixxxi.).
(vfel" 1 • • • 1 • ' * L '^^'' V''V • • e*<"'+'^+ . . . +x^+ . . . ^.^dxidx^. ..dx,...dx„
= H,H, . . . H, . . . H^A . . . i8, . . . /8/%i4=i4^ . . . ,
Ps Pr Pt"
or
j j . . . j . . . j ZqU{,v^) dx^dxc^ . . . dx, . . . dxn = QqH r^)
. . . . (Ixxxii.).
where 11 denotes a product of ,Vj, for any number of v's with any s and p. The rule,
therefore, is very simple. We must expand the value of z ini/& as given by (Ixxviii.)
above, then the multiple integral of this will be obtained by lowering every v's right
hand subscript by unity (remembering that sVq =1), and further dividing by the fi of
the lefthand subscript. The general expression up to terms of the fourth order has
been written down ; it involves thirtyfour siuns, each represented by a type term
All these would only occur in the case of the correlation of eight organs, or when we
have to deal with twentyeight coefficients of correlation. Such a number seems
beyond our present power of arithmetical manipulation, so that T have not printed the
general expressions. At the same time, the theory of multiple correlation is of such
great importance for problems of evolution, in which over and over again we have
three or four correlated characters to deal with,'*^ that it seems desirable to place
on record the expansion for these cases. I give four variables up to the fourth and
three variables up to the fifth order terms. Afterwards I will consider special cases.
* In my memoir on Prehistoric Stature I have dealt with five correlated organs, *.«., ten coefficients. In
some barometric investigations now in hand we propose to deal with at least fifteen coefficients, while
Mr. BramleyMoore, in the correlation of parts of the skeleton, has, in a memoir not yet published, dealt
with between forty and fifty cases of four variables or six coefficients.
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Vcdue of the Quadruple Integral in the Case of Four Variables.^
Qo fil^% A^3 ^1^4 ^3^3 ^2^4 ^8^4
^1^3^8 ^ ^1^2^3 ^ A^2^3^4 A^2^3^4
^^18^*84 //' I ^!li^J4 ,, iv _j_ ^^'28^'8.
^ ^^18^^84 ^ /// _^ fhj^u t; >^ + fVM <y '
+
^^*24^84 «i iv I ^^12^*24 «. " ^_ ^^18^*24
^2^3^4 ^ AA^4 ^ ^i^2^3^4
i 2^14^24 iv I ^^28^24 //I
+/3i/32/34 ''^ + fiJ3^, ""' J
+ 1^1/31/8/^''^ ^ Ms'"'''' +/8i/8/^^^ +/8a''^'^ +/92/8/^^^
+ /3A '* ''^ + M^ '''''' + Ay8A ''^''^ + /^^ ''^''^
^^l^^2^>3 P1P2P8 P1P2P3
3^13?^ ,„ „ 3r^/ro3 , .^, ^^'u^'23^ " ///
+ A/8^A ''^ "^ + /9i/9^. ''^ "' + A/3A "^ '''
* To simplify the notation, »,', »,", «;,'" »,'» have been used for i»„ jf,, 8»«, 4»»
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30 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS
+ /8A/9A ^ ^ +/3i/S,/93^4 ^ > +/8i/8^A ^^
6rigrggr34 , ,„ 6r)3r33^^^ ,„ 6ri4rg8r^ ,„ .^
+ )8,^A ^ ^ ^ + /3i;SA ''^''^ "^^ J
+ liW, " * +i8ii9s '''» +A/8/'' ' +/3W38 * ' +.8^4'*^ ^/Sa'^''^^
^ i8i/8^8 "^^ '■' ^ iSi^A "^^ ' + ^A^4 "^^ ' ^ /e./8A/84 ''^ ''«
+ /Sii8,i94 * '^ + i8>^A ' '^ + /8i/3w38^« "^'"^^ + fi^M^. '' ^'
+ /3i/9A ^^ + i8i/8A ^ '^ ^ /3^A " * ^ /9,/88i8. " *
, ^^'l8 ^34 / // , ^ 4?j£s/_ „, j^ ^^'l8^ ^SA /// / i ^^'l8^84^ /// jy
, ^14' ^84 / iy , ^^uV „ '// w , 4rj/»^ „ „ 4rg3»^ ,„ j^
+ /Si/SsS, ^«^« + A/8A ' ' + /8^A ' ' ^ M^* ' '
+ ^AA "' "' + ^. "' "• + ^ "• '
. 6r„
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I ^^^'la ^14^*24 f ft \v % *■ ^^12^14 ^'24 / // iv I "'••^^12^14^91 / // ;»
+ A)8A ^'^^^ ^» + A/8A ^ ' ^ ^ A/S3/S, ^'»^* ^«
IV
+ A/SW3A ^ » * + /3i/8^A ' ' + /3i/3^A ^ ' ^ »
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32 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS
^ /Ss/8A '''' ""^ ""' + /8^,/3, *^ ""' ''« + /9^,/3, ''l ''^ ''^
, 2^12 ^18^14^23 / // m , 24713 ^13^1^ , „ ;^ 24rigri3rayi4 / ^/ //'
^ /8i/32)8A "^^"^^"^^ + /8,/82/33/8, ^'^^^ ^^ ^ ^1/8,^3/3, ^^ "« '^^
+ i9ii82/9A ^ '^ ^ + /9i)82/9A ^1^1^^ ^1 + /81/3,^3/S, ^«^i ^1
+ /3i;8,^3/3, ^^^i"« + /9i/33)S3/3, ^^^^^1 ^1 + /3i;8,/33/9, ^'^ ^* ^^
+ A/S^a^* ""^'^ ""' + Ay92/9A ^^^1^^ + fi^^^, ^1^1 ^'^
. . . . (Ixxxiii.).
In the case of three variables, we must cancel in the above expression all terms
involving jS.^ Thus we shall have 3 instead of 6 first order terms, 6 instead of 21
second order terms, 10 instead of 56 third order terms, and 15 instead of 126 fourth
order terms — a much more manageable series.
I give below the extra term necessary for calculating the value of (Q — Qo)/Qo a.s
far as the fifth order terms in the case of three variables.
Fifth Order Terms for Three Vanables.
PiPiP% PiPA PiPoPa
ins** iHtS*"^ lO'S'*'*
+
20r.
f/t
+ ^^A'"^^^''^« + /3./3A '^^^''^ I ^^''''''^'^•^•
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A numerical illustration of these formulsB will be given in the latter part of this
Memoir. It will, however, be clear that what we want are tables of log ( ^ j, including
log ('^) or log [—1 for a series of values of h. Such tables would render the compu
tation of — jr^ fairly direct and rapid ; they could be fairly easily calculated from
existing tables for the ordinate and area of the normal curve, and I hope later to
find some one willing to undertake them.
Meanwhile let us look at special cases. In the first place, suppose, in the case of
three variables, that the division of the groups is taken at the mean, ?.e., ^^ = A^ =
A3 = 0. Then we have
/8i = A = )83 = fV*''<^=»=A/
v/ = v{ = v/" =
< = < = <'= 1
< = v^' = <' =
< = < = <" = 3.
Hence we have
= Qojl + — (sinirij + sin'ris + sin^rja)! (Ixxxv.).
Let ri2 = cos D^, rjg = cos D^g, r^ = cos Djg, and let E be the spherical excess of
the spherical triangle whose angles are the divergences D^, Djj, D23. Then
we have
Q — Qo ^ _ ?E T\ r» r\ —.'^
Qo 2  2 ~ ^i» ~ ^^3 ~ ^«8  2
Or: sin^^J=cosE (Ixxxvi.).
Now take the case of four variables. Here we have
^1 = ^2 = ^3 = ^4 = V 5
< = V = <" = 1Jji» = 1
and all the odd v's zero. Hence
VOL. cxcv. — ^A. p
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34 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS
Q 2' / 2 \®
— Q— ^ = ;. ('12 + ''js + »*it + ^23 + »'24 + r^) + [) ('•w'as + rijra, + r^^r^^)
*> 1 / 2 \®
+ ^ ]3 (**•«' + ''is" + '■"' + ''2S' + ^2*' + **»*') + (^ j (*'18'*13'*14 + ^12*
/2 V 1
+ 'Isna'M + »*]*''24»*84) + \^y i3"(^l*'*'28 + '•l4»'23' + ^3^24* + ^3% + ^llV,^
/ 2 \^ 1
+ ^12^3*^) + [yj 12" (^12^^14^23 + ^Au'^SS + ^12^''l3^54 + ^18^'u^^24 + ^13^23^^24
+ ^4^23^ + ^2^13^4 + ^12^^34 + ^2^23^^*34 + ^14^28^34^ + ^2^24^^^
+ ^13^24^34^) + (IxXXvii.).
This is the correct value including terms of the fourth order, but to this order of
approximation we can throw it into a much simpler form. Let r,y = sin 8,^, then
— p— ^ ^ = sin"^ rjg + sin"^ rjg + sin~^ r^^ + sin"^ Vc^ + sin"^ r^^ + sin~^ r^
2 .
+ — (sin"^ 7*13 sin""^ r^^ sin""^ r^^ + sin"^ r^^ sin"^ 7\2Z ^^^"^ ^24
+ sin~^ r^g sin~^ r^^ sin""^ r^ + sin""^ r^^ sin"^ r^^ sin"^ r^^)
+ ~ [sin» n* sin^ r«,{(l  r,,«) (1  n,') (1  r^») (1  r,,*)]*
+ sini ri^sini r3,{(l  r,,^) (1  r,,*) (1  r^') (1  V)}"*
+ sini r,3 sini r,,{(l  V) (1  r,,«) (1  V) (1  rg/)}"*]
= 8j3 + 8i3 + 8j4 + 823 + 824 + ^34
+ ^ (812813^14 + 813833824 + 813823834 + 814824834)
2^ A^ggg COS 8^ COS 8g3 4 8128^ COB 812 COS 83^ 4 SigSg^cofr 8,^00382 A
^ \ COS 813 COS Si3 cos Si4 COS Sjj cos 834 COS Sjj4 /
. . . • (Ixxxviii.).
We may write this sin — ^— ^ ^ = cosE' (Ixxxix.)
where
E' = "2 "■ 812 — 813 — 814 — 823 — 824 — 834
— ^ (812833814 + 810823834 + 813823824 + 814824834)
2 / 8i^833 cos 834 cos 823 f 8,38^005812008834 4 813824 cos 8i3 COB 83t \
IT \ COB 8j3 COS Si3 COS 834 COS 853 COS 834 COS 834 /
The expressions E and E' of (Ixxxvi.) and (Ixxxix.) are of considerable interest, for
they enable us to express the area of a spherical triangle in threedimensioned space,
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35
and (up to the above degree of approximation) the volume of a " tetrahedron " on a
" sphere " in hyperspace of four dimensions. In fact, the whole theory of hyperspace
" spherical trigonometry " needs investigation in relation to the properties of multiple
correlation.
In our illustrations (viii.) and (ix.) will be found examples of the above formulae
applied to important cases in triple and quadruple correlation in the theory
of heredity. I consider that the formulae above given will cover numerous novel
applications, for many of which greater simplicity will be introduced owing to the
choice of special values for the h's or for the correlation coefficients.
(8.) Illustrations of the New Methods.
Illustration I. Inheritance of Coat colour in Horses. — The following represents
the distribution of sires and fillies in 1050 cases of thoroughbred racehorses, the
grouping being made into aU coatcolour classed as ** bay and darker," " chesnut and
lighter":—
Colour.
Sires.
Bay and
darker.
Chesnut and
lighter.
FiUiea.
Bay and darker . . .
631
125
756
Chesnut and lighter .
147
147
294
778
272
1050
a
h
a + b
c
d
c + d
a + c
b + d
N
Then we require the correlation between sire and filly in the matter of coatcolour,
and also the probable error of its determination.
We have from (iv.) and (v.)
«i = ^ \^ ^ = \/ \e^^dx = •481.905,
«2 = ^ V^ ^ = V ^Jo^ '^^y = 440,000.
Hence from the probability integral tables
h = 64630, k = 58284.
We have then : log HK = T037,3514 by (xvii.),
F 2
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36 PROFESSOE K. PEARSON ON MATHEMATICAL CONTRIBUTIONS
Thence c = "'~J = 619,068 from (xxi.).
Calculating out the coefficients of the series in r in (xix.) we find
•619,068 = r + •188,345r2 + •064,0814r« + •107,8220r* + '005,99867^ + •067,2682r«
4 &c.
Neglecting powers of r above the second, we find by solving the quadratic and
taking the positive root
r — 5600.
Solving by two approximations the sextic we finally determine
. r = 5422,
correct, I think, to four places of figures.
Turning now to the probable error as given by Equation (1.), I find
^s + jfc2 _ 2rhk = 348,924,
and from (xlix.)
logxo= 1*170,0947.
Further : '^ll\^ ~ 275,642 , Jpi^ = '393,078.
Hence from (xlvii.) and (xlviii.) we find
1 r»93,078 1 r 275,642
and by means of the probability integral table
iri = 108,884, xji^ = 152,865.
By substituting in (1.), we find
probable error of r = '0288.
From (xxxiv.) and (xxxv.) we find
p.e. of A = 0282. p.e. oi k = 0278.
Thus, finally, we may sum up our results
h = 6463 ± 0282, k = '5828 ± '0278,
r = 5422 ± 0288.
The probable enor of this r, if we had been able to find it from the product
moment, would have been '0147, or only about onehalf its present value.
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Illustration IL — Our analysis opens a large field suggested by the following
problem : — What is the chance that an exceptional man is born of an exceptional
father f
Of course much depends on how we define " exceptional," and any numerical
measure of it must be quite arbitrary. As an illustration, let us take a man who
possesses a character only possessed by one man in twenty as exceptional. For
example, only one man in twenty is more than 6 feet 1'2 inches in height, and such a
stature may be considered " exceptional." In a class of twenty students we generally
find one of " exceptional " ability, and so on. Accordingly we have classed fathers and
sons who possess characters only possessed by one man in twenty as exceptional. We
first determine h and Jc, so that the tail of the frequency curve cut off is ^ of its
whole area. This gives na h = k = 1*64485.
Next we determine HK = ^e'^'''^'*\ and find log HK = 2026,8228.
Then we calculate the coefficients of the various powers of 7* in (xix.). We find
\og^hk= 131,2225.
logi(/t«  1) (F  1) = 1685,5683.
log^Ah^  3)(^  3) = 3990,1176.
logxio (A*  6/i* + 3)(**  6F + 3) = 1464,4772.
log^(A*  10A« + 15)(**  lOP + 15) = 29.25,6367.
It remains to determine what value we shall give to r, the paternal correlation. It
ranges from "3 to "5 for my own measurements as we turn from blended to exclusive
inheritance. Taking these two extreme values we find
— ^r=r = 0046344 or '0096779.
But — 2 = — — ^ :^ ^ and the second term is the chance of exceptional
fathers with exceptional sons, when variation is independent, i*e., when there is no
heredity, = ^ X Vo = '0025.
Thus d/N = 007134 or 012178 ;
accordingly 6/N = 042866 or 037822.
Hence we conclude that of the 5 per cent, of exceptional men 71 per cent, in the
first case, and 1*22 per cent, in the second case, are bom of exceptional fathers, and
429 per cent, in the first case and 378 per cent, in the second case of nonexceptional
lathers. In other words, out of 1000 men of mark we may expect 142 in the first case.
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38
PEOFESSOR K. PEAESON ON MATHEMATICAL CONTRIBUTIONS
244 in the second, to be born of exceptional parents, while 858 in the first and 756 in
the second are born of undistinguished fathers. In the former case the odds are about
6 to 1, in the latter 3 to 1 against a distinguished son having a distinguished father.
This result confirms what I have elsewhere stated, that we trust to the great mass of
our population for the bulk of our distinguished men. On the other hand it does not
invalidate what I have written on the importance of creating good stock, for a good
stock means a bias largely above that due to an exceptional father alone.
In addition to this the •^ of the population forming the exceptional fathers pro
duce 142 or 244 exceptional sons to compare with the 858 or 756 exceptional sons
produced by the ^ of the population who are nonexceptional. That is to say that
the relative production is as 142 to 45*2, or 244 to 39*8, i.e., in the one case as more
than 3 to 1, in the other case as more than 6 to 1. In other wordsy exceptional
fathers produce exceptional sons at a rate 3 to 6 times as great as nonexceptional
fathers. It is only because exceptional fathers are themselves so rare that we must
trust for the bulk of our distinguished men to the nonexceptional class.
Illustration III. Heredity in Coatcolour of Hounds. — To find the correlation
in coatcolour between Basset hounds which are halfbrethren, say, ofispring of the
same dam.
Here the classification is simply into lemon and white ijiiv) and lemon, black and
white or tricolour (f),
The following is the table for 4172 cases : —
Colour.
t.
IV).
Totals.
t.
1766
842
2608
Iw.
842
722
1564
Totals
2608
1564
4172
Proceeding precisely in the same way as in the first illustration we find
«! = Oj =
h=zh =
25024
318,957
157,6378
logKH=:l
c= 226,234.
It will be suflficient now to go to r*. We have
•226,234 = r + '050,867 r* + 134,480 r» + '035,587 v\
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The quadratic gives r = '2237. Using the Newtonian method of approximating to
the root we find
r = 2222.
Summing up as before, after finding the probable errors, we have
h = k= 3190 ± 0133,
r= 2222 ±0162.
Illustration IV, Inheritance of Eyecolotir in Man.— To find the correlation in
eyecolour between a maternal grandmother and her granddaughter. Here the
classification is into eyes described as grey or lighter, and eyes described as dark grey
or darker.*
Tint.
Maternal grandmother.
Totals.
Grey or lighter.
Dark grey or
darker.
1
1
1
C5
Grey or lighter ....
254
136
390
Dark grey or darker . ,
156
193
349
Totals
410
329
739
As before, we find
aj = 109,607, ajj = 055,480,
h = 138,105, k = 069,593,
log HK = T196,6267,
c = 323,760.
Series for r up to r*
•323,760 = r + 004,806?^ + 162696r5 + '000,358^*
The quadratic gives r = "3233, and the biquadratic
r = 3180,
the value of the term in r* being 000,00366, so that higher terms may be neglected.
Determining the probable errors as in Illustration I., we sum up : —
* According to Mr. Galton's classification, the first group contains eyes described as light blue, blue,
dark blue, bluegreen, grey ; and the second eyes described as dark grey, hazel, light brown, brown, dark
brown, verv dark brown, black.
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40 PKOFESSOE K. PEAESON ON MATHEMATICAL CONTBIBUTIONS
h = 1381 ± 0312,
k = 0696 db 'OSll,
r = 3180 ± 0361.
Illustration V. Inheritance of Stature. — The following data have been found for
the inheritance of stature between father and son from my Family Data cards, 1078
cases : —
Mean stature of father. . . . 67"698
son 68''661
Standard deviation of father . . 2"7048
son. . . 2"7321
Correlation = 5198 ± '0150.
Now for purposes of comparison of methods the correlation has been determined
for this material from various groupings of fathers and sons : —
(A.)
Fathers.
«
Class.
Below 67"5.
Above 67"5.
Totals.
Below 67"6 . .
26925
9675
365
Above 67"5. .
23225
48075
713
Totak . . .
5016
5765
1078
(B.)
Fathers,
CQ
Class.
Below 66"5.
Above 66"5.
Totals.
Below 67"5 . .
21125
15375
365
Above 67"5 . .
15275
56025
713
Totals . . .
364
714
1078
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TO THE THEORY OF EVOLUTION.
Fathers,
41
Class.
Below 67"5.
Above 67"5.
Totals.
Below 686" . .
35626
18225
6385
Above 68'5" . .
14625
39425
5396
Totals . . .
5016
6766
1078
(D.)
Fathers.
CQ
CSass.
Below 68"6.
Above 68"5.
Totals.
Below 69"6 , .
506
182
688
Above 69"5 . .
1495
2406
390
Totals . . .
6556
4226
1078
(E.)
Fathers.
Class.
Below 69"5.
Above 69"6.
Totals.
Below 70"5 . .
669
147
816
Above 70"6 . .
128
134
262
Totals . . .
797
281
1078
(F.)
«
Fathers,
Class.
Below 70"5.
Above 70"5.
Totals.
Below 69"5 . .
64125
4676
688
Above 69"5 . .
27175
11825
390
Totals . . .
913
165
1078
VOL. OXCV. — A.
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42
PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS
Table of Results.
Claasificatiom.
Correlation.
Mean of sons.
Mean of fathers.
A
B
D
E
F
•5939 ± 0247
■5557 ± 0261
•5529 ± 0247
5264 ± 0264
•5213 ± 0294
5524 ± 0307
k.
68"64( 416,32)
68"64( 416,32)
68"50( 001,16)
68"53 (353,71)
68"60 (696,57)
68"63 (353,71)
A.
67"74( 087,00)
67"63( 418,86)
67"74( 087,30)
67"77 (274,30)
67"76 (641,30)
67"73 (1023,44)
Now these results are of quite peculiar interest They show us : —
(i.) That the probable error of r, as found by the present method, increases with
h and k But the increase is not very rapid, so that the probable errors of the series
range only between '025 and •031. Hence while it is an advantage, it is not a very
great advantage, to take the divisions of the groups near the medians. It is an
advantage which may be easily counterbalanced by some practical gain in the method
of observation when the division is not close to the medians.
(ii.) While the probable error, as found from the present method of calculation, is
1*5 to 2 times the probable error as found from the product moment, it is by no
means so large as to seriously weigh against the new process, if the old is un
available. It is quite true that the results given by the present process for six
arbitrary divisions diflPer very considerably among themselves. But a consideration
, of the probable errors shows that the differences are sensibly larger than the prob
; able error of the differences, even in some case double ; hence it is not the method
, but the assumption of normal correlation for such distributions which is at fault. As
' we shall hardly get a better variable than stature to hypothesise normality for, we
see the weakness of the position which assumes without qualification the generality
of the Gaussian law of frequency.
(iii.) We cannot assert that the smaller the probable error the more nearly will
the correlation, as given by the present process, agree with its value as found by
the product moment. If we did we should discard '5213, a very accordant result,
in favour of '5529, or even '5939. The fact is that the higher the correlation the
lower, ceteris "paribus^ the probable error, and this fact may obscure the really best
result. Judging by the smallness of h and k and of the probable error, we should
be inclined to select C or the value '5529. This only differs from '5198 by slightly
more than the probable error of the difference ('033 as compared with '029) ; but
since both are found from the same statistics, and not from different samplings ot
the same population, this forms sufficient evidence in itself of want of normality.
The approximate character of all results based on the theory of normal frequency
must be carefully borne in mind ; and all we ought to conclude from the present
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data for inheritance of stature from father to son would be that the correlation
= '55 ± '015, while the product moment method would tell us more definitely that
its value was '52 ± 015. There is no question that the latter method is the better,
but this does not hinder the new method from being extremely serviceable; for
many cases it is the only one available.
Illustration VI. Effectiveness of Vaccination. — To find the correlation between
strength to resist smallpox and the degree of eflfective vaccination.
We have in the earlier illustrations chosen cases in which in all probability a scale
of character might possibly, if with difficulty, be determined. In the present case,
the relationship is a very important one, but a quantitative scale is hardly discover
able. Nevertheless, it is of great interest to consider what results flow from the
application of our method. We may consider our two characters as strength to resist
the ravages of smallpox and as degree of efiective vaccination. No quantitative
scales are here available ; all the statistics provide are the number of recoveries
and deaths from smallpox, and the absence or presence of a definite vaccination
cicatrix. Taking the Metropolitan Asylums Board statistics for the epidemic of 1893,
we have the table given below, where the cases of " no evidence " have been omitted.
Proceeding in the usual manner we find
a^= 86929 a^ = 54157
h= 151139 *= 74145
c = 782454.
Hence the equation for r is
•782,454 = r + '560,310?'^  096,3787^ + 081,8817^  •000,172?'^  •040,0597'^^
whence r = '5954.
Simiming up we have, after calculating the probable errors,
h= 15114 ± 0287,
k = 7414 ± 0205,
r = 5954 db 0272.
Strength to resist Smallpox when incurred.
g g
Cicatrix.
Becoveries.
Deaths.
Total.
Present
1562
42
1604
Absent
383
94
477
Total
1945
136
2081
O 2
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44
PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS
We see accordingly that there is quite a large correlation between recovery and the
presence of the cicatrix. The two things are about as closely related as a child to its
" midparent." While the correlation is very substantial and indicates the protective
character of vaccination, even after smallpox is incurred, it is, perhaps, smaller than
some over ardent supporters of vaccination would have led us to believe.
Illustration VIL Effectiveness of Antitoxin Treatment. — ^To measure quanti
tatively the effect of antitoxin in diphtheria cases.
In like manner we may find the correlation between recovery and the administration
of antitoxin in diphtheria cases. The statistics here are, however, somewhat diflScult
to obtain in a form suited to our purpose. The treatment by antitoxin began in the
Metropolitan Asylums Board hospitals in 1895, but the serum was then administered
only in those cases which gave rise to anxiety. Hence we cannot correlate recovery
and death with the cases treated or not treated in that year, for those who were likely
to recover were not dosed. In the year 1896 the majority of the cases were, on the
contrary, treated with antitoxin, and those not treated were the slight cases of very
small risk ; hence, again, we are in great difficulties in drawing up a table.* Further,
if we compare an antitoxin year with a nonantitoxin year, we ought to compare the
cases treated with antitoxin in the former year with those which would probably have
been treated with it in the latter year. Lastly, the dosage, nature of cases treated,
and time of treatment have been modified by the experience gained, so that it seems
impossible to club a number of years together, and so obtain a satisfactorily wide
range of statistics. In 1897, practically aU the laryngeal cases were treated with
antitoxin. Hence the best we can do is to compare the laryngeal cases in two years,
one before and one after the introduction of antitoxin. The numbers available are
thus rather few, but will help us to form some idea of the correlation. I take the
following data from p. 8 of the Metropolitan Asylums Board * Report upon the Use of
Antitoxic Serum for 1896 ' : —
Laryngeal cases.
Eecoveries.
Deaths.
Totals.
With antitoxin, 1896 ....
319
143
462
Without antitoxin, 1894 . . .
177
289
466
Totals
496
432
928
* When a new drug or process is introduced the medical profession are naturally anxious to give every
patient the possible benefit of it, and patients of coiu*se rush to those who first adopt it. But if the real
efficiency of the process or drug is to be measured this is very undesirable. No definite data by which to
measure the effectiveness of the novelty are thus available.
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Here I find r = '4708 ± '0292.
A further table is of interest : —
Laryngeal cases.
Kequiring
tracheotomy.
Not
requiring it.
Totals.
Without antitoxin, 1894 . . .
261
205
466
With antitoxin, 1896 ....
188
274
462
Totals
449
479
928
In this case we have r = "2385 ± 0335.
Lastly, I have drawn up a third table :—
Total Infantile Cases, Ages — 5 years.
Recovery.
Death.
Totals.
With antitoxin, 1896 . . .
912
434
1346
Without antitoirin, 1894 . .
615
556
1171
Totals
1527
990
2517
Here we have* r = '2451 ± 0205.
The three coeflficients are all sensible as compared with their probable errors, and
that between the administration of antitoxin and recovery in laryngeal cases is
substantial. But the relationship is by no means so great as in the case of vaccina
tion, and if its magnitude justifies the use of antitoxin, even when balanced against
other ills which may follow in its train, it does not justify the sweeping statements of
its eflfectiveness which I have heard made by medical ifriends. It seems until wider
statistics are forthcoming a case for cautiously feeling the way forward rather than for
hasty generalisations.
Illustration VII f. Effect on Produce of Superior Stock. — To find the eflfect of
superiority of stock on percentage goodness of produce.
To illustrate this and also the formula (Ixxxiii.) for six correlation coefficients, we wiU
investigate the eflfect of selecting sire, dam, and one grandsire on the produce when there
* The values of r for all the three cases of this Illustration were determined with great ease from
Equation (xxiv.).
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46 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS
is selective pairing of dam and sire. We will suppose grandsire, dam, and sire to be
above the average, and investigate what propoition of the produce will be above the
average. As numbers very like those actually occurring in the case of dogs, horses,
and even men, we may take
Correlation of grandsire and offspring . = '25
„ sire or dam and offspring = "5 in both cases
„ sire and grandsire . . . = "5
Selective mating for sire and dam. . . = "2
We will suppose zero correlation between paternal grandsire and dam, although
with selective mating this may actually exist.* We have then the following
system : —
^14? ^^ 25, r^ = '5, r^^ = '5, t^ = '2, Vi^ := '5, r^^ = 0.
Hence, substituting these values in (Ixxxvii.), we find — ^after some arithmetic :
(QQo)/Qo= 14851.
But Qo is the chance of produce above the average if there were no heredity
between grandsire, sire, and dam, and no assortative mating.
N
Hence it equals iX^X^X^N^— .. Q= '1553 N.
Or, of the produce '5 N above the average, '1553 N instead of '0625 N are bom of
the superior stock owing to inheritance, &c. In other words, out of the '5 N above
the average, '1553 N are produced by the stock in sire, dam, and grandsire above the
average, or by '1827 of the total stock, t The remaining '8173 only produce '3447 N,
or the superior stock produces produce above the average at over twice the rate of the
inferior stock. Absolutely, the inferior stock being seven times as numerous produces
about seventenths of the superior offspring.
Illustration IX, Effect of Exceptional Parentage. — Chance of an exceptional
man being bom of exceptional parents.
Let us enlarge the example in Illustration II., and seek the proportion of exceptional
men, defined as one in twenty, born of exceptional parents in a community with
assortative mating.
* A correlation, if there be substantial selective mating, may exist between a man and his motherin
law. Its rumoured absence, if established scientifically, would not, however, prove the nonexistence of
selective mating, for A may be correlated with B and C, but these not correlated with each other.
t The proportion of pairs of parents associated with a grandsire above the average was found by
putting 5, % and for the three correlation coefficients in (Ixxxv.). In comparing with Illustration II.,
the reader must remember we there deiilt with an exceptional father, 1 in 20, here only vdth relatives
above the average — a very less stringent selectio^Fi.
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TO THE THEOEY OF EVOLUTION. 47
Here we taJce for father and son r^^ = "5, for mother and son i\^ = '5, and for
assortative mating, r^^ = '2.
We have then to apply the general formulae (Ixxxiii.) and (Ixxxiv.) for the case of
three variables. We have
Ai =^2 =^3 = 164485
A = ^9 = iSa = 484,795
Ui' = V = V" = 1 '644,850
Vj' = V," =: t>g"' = 1705,532
v,' = <' = V = — 484,356
v^' = <' = V' =  5913,290
Whence, after some arithmetical reduction, we find
(Q  QoVQo = 200389.
ButQo = ^X^X^N = TT^tf^ N. Hence Q = 00263 N.
We m\ist now distinguish between the absolute and jllative production of excep
tional men by exceptional and nonexceptional parents. The exceptional pairs of
parents are obtained by (xix.), whence we deduce, putting r = 2, ^ = i = 1*64485,
'^^ _ A (d + b)(d + c)_ d 1 _ .o^oy, .
Whence the number of pairs of parents, both exceptional
= 005245 N.
Thus, '005245 N pairs of exceptional parents produce '00263 N exceptional sons,
and '994755 N pairs of parents, nonexceptional in character, produce '04737 N
exceptional sons, i.e., the remainder of the ^ N. The rates of production are thus as
'5014 to '0476. Or : Pairs of exceptional parents produce exceptional sons at a rate
more than ten times as great as pairs of nonexceptional parents. At the same time,
eighteen times as many exceptional sons are bom to nonexceptional as to exceptional
parents, for the latter form only about J per cent, of the community.
The reader who will carefully investigate Illustrations II., VIII., and IX. will grasp
fully why so many famous men are born of undistinguished parents, but will, at the
same time, realise the overwhelming advantage of coming of a good stock.
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[ 49 ]
II. Electncal Conductivity in Gases Traversed by CatJwde Rays.
By J. C. McLennan, Demonstrator in Physics^ University of Toronto.
Communicated by Professor 5. J. Thomson, F.R.S.
Received December 7, 1899— Read February 1, 1900.
Though it has been known that a gas becomes a conductor when traversed by cathode
rays, yet the laws connecting this electrical conductivity have not hitherto been
studied.
The theory has been put forward by J. J. Thomson and Rutherford* that when a
gas becomes a conductor under a radiation, it does so in virtue of the production of
positive and negative ions throughout its mass. This view has been established by
their experiments on Rontgenised gases, and confirmed by those of ZELENYf on the
same subject. The recent work of Rutherford on Uranium Radiation^ also affords
another example of such a process in the gases traversed.
The object of the experiments which are described in this paper was to investigate
the nature of the conductivity in different gases when cathode rays of definite
strength passed through them, and to measure the number of ions produced. With
this in view, I have worked with cathode rays produced, after the method of Lenard,
outside the discharge tube, as these were found to be more easily dealt with than
those inside.
The investigation is described under the following subdivisions : —
1. Form of tube adopted for the production of cathode rays.
2. Ionization by cathode rays.
3. Discharging action of cathode rays.
4. Ionization not due to Rontgen rays.
5. Discussion of methods for measuring the ionizations produced in different
(es.
6. Description of apparatus used.
7. Explanation of the method adopted for comparing ionizations.
8. Ionization in different gases at the same pressure.
9. Ionization in air at different pressures.
♦ * Phil. Mag.,' November, 1896, p. 393.
t *Phil. Mag.,' July, 1898, p. 120.
X *Phil. Mag.,' January, 1899, p. 109.
VOL. CXCV.— A 263. H 3.11.1900.
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MR J. c. Mclennan on electrical conductivity in gases
10. Ionization in a gas independent of its chemical composition.
11. Comparison of ionizations produced by cathode and by Rontgen mys.
12. Summary of results.
1. Form of Tube adopted for the production of Cathode Rays.
To produce the rays, a modified form of the tube devised by Lenard,* fig. 1, was
used. The disc a which closed the end and carried the aluminium window formed
the anode. To hold this disc in position, and to render the joint airtight, recourse
was had to sealingwax, which was allowed to set on the previously warmed glass and
metal, after which the parts were made to unite by slightly melting the surfaces and
pressing them together. By running round the joint with the pointed flame of a
blowpipe, any air bubbles present were removed, and complete union was effected.
Joints made in this way were found to hold for any time desired.
In making the aluminium window airtight, marine glue could be used, but the
ordinary commercial soft wax was found to be more suitable. This was especially so
when the experiments were in the tentative state and alterations were fi'equently
necessary. The wax melted at a lower temperature than the glue, and besides being
much more manageable than the latter, it was also less disagreeable to handle. A
coating of it on the sealing wax also prevented cracking.
As shown in the figure, the anode was provided with a shoulder round the opening
of the window. This was found very convenient when the action of the rays on the
J^i^fi
air in a partially exhausted receiver such as A was 'oeing examined. The receiver
was provided with a similar but larger shoulder, and by slipping it over that on the
* * Wied. Ann.,' vol. 51, p. 225 (1894).
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TRAVERSED BY CATHODE RAYS. 51
anode and applying a coating of wax, an airtight connection could be readily made
without interfering with that which secured the aluminium foil to the disc. This
latter connection was effected by placing a thin coating of wax upon the brass disc
and gently applying heat after the foil was laid upon it. All the space within the
projecting shoulder was then covered with a thick coating of the wax, excepting the
central portion of the aluminium.
In all the experiments with these tubes the anode was well earthed, as was also the
positive terminal of the induction coil used to produce the discharge.
As regards the distance between the cathode and the anode, it was found best not
to make it too small. Otherwise, the discharge would pass in the tube before the
available maximum potential difference was reached. The velocity of the carriers has
been shown by J. J. Thomson* to vary with the potential difference between the
electrodes, and as a consequence an intense radiation was more readily obtained when
the distance between the anode and cathode was considerable.
In the case of tubes constructed with a short distance between the electrodes, the
device adopted by McCLELLANDt of inserting an air gap in series with the tube very
largely increased the intensity of the radiation.
The foil used by Lenard for the aluminium window was '003 millim. in thickness.
In practice it was exceedingly difficult to obtain such foil free from holes. Aluminium
about three times as thick was, however, much better in this regard. The induction
coil used in the experiments was, besides, very powerful, and, as a radiation
sufficiently intense could be obtained with it, this thickness was used throughout the
investigation.
2. Ionization by CatJwde Rays.
It has been shown by Lenard that air, when traversed by cathode rays, acquires
the property of discharging electrified conductors against which it may be blown, and
that, fiirther, it retains this property for some time after the rays producing it have
been cut off.
According to the theory of Professor Thomson, the air, when in this state, is
ionized, and the discharging action is brought about by a motion of the ions in the
gas to the charged conductor. Owing to the separation of the positive and negative
ions, recombination can take place but gradually, and this readily explains why the
discharging power is retained by the air for some time. In order to show that these
positive and negative ions are produced in a gas traversed by the rays, the apparatus
shown in fig. 1 was used.
The cathode rays issuing from the aluminium window a passed through a narrow
tube, 6, into an earthconnected metal chamber, A. B was a disc of brass supported
♦ *Phil. Mag.,' October, 1897, p. 315.
t *Proc. Roy. Soc.,' vol. 61, No. 373, p. 227.
X * Wied. Ann.,' vol. 63, p. 253 (1897).
H 2
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52 MR. J. c. McLennan on electrical conductivity in gases
by an ebonite plug, and surrounded by a guard ring. A wire led from this electrode
to one pair of quadrants of an electrometer, and the other pair was put to eaith.
Care was taken to screen off electrostatic induction by surrounding the wire and
electrometer with earthconnected conductors. The second electrode, C, also
supported by an ebonite plug, was connected by a commutator, D, to one of the
terminals of a battery of small storage cells, the other terminal being connected to
earth.
The tube, 6, was made narrow, and penetrated a short distance into the chamber in
oixler to confine the rays to a slender pencil, and to prevent their impinging upon the
electrodes. By means of the key, K, the electrode, B, could be put to earth when
necessary.
With such an apparatus, and no field initially between the electrodes, it was found
on exciting the discharge tube and breaking the earth connection, K, that the
electrometer gained a small negative charge, which did not go on increasing, but soon
attained a limiting value.
On the assumption that the cathode rays produce positive and negative ions
throughout the gas, the explanation of this is obvious. The cathode rays carried a
negative charge into the gas, and set up a field which caused the negative ions to
move to the walls of the chamber and to the electrode, B. The charge which the
latter soon gained, however, set up a field of its own, and a state of equilibrium was
reached when the conduction to the electrode was just equal to that proceeding from
it. If, instead of there being no field initially between the electrodes, C was joined
to the positive terminal of the batteiy, then the electrode, B, gained a positive charge
when the tube was excited, and the rate at which its potential rose depended upon
the capacity joined to B and the electrometer.
With C joined to the negative terminal of the battery, a similar charging took
place, except that in this case the charge accumulated was a negative one.
This reversal in the sign of the charge collected may be shown with a field of a
few volts a centimetre, and clearly points to the existence of positive and negative
ions in the gas. Since the cathode rays themselves carry a negative charge, the
presence of these carriers alone in the chamber would account for the negative charge
obtained with a negative field. With a positive field, however, these carriers would
be attracted to the electrode C, and it seems impossible to explain how the electrode B,
under these circumstances, could receive a positive charge unless ions were produced
by the rays.
3. Dischargivff Action of Cathode Rays,
In connection with the experiments of Lenard,* already referred to, cathode rays
were allowed to fall upon a charged conductor surrounded with air at atmospheric
pressure. This conductor consisted of a wire attached to a goldleaf electroscoj^e,
♦ • Wied. Ann.,' vol. 63, p. 253.
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TRAVERSED BY CATHODE RAYS. 53
and was placed within a zinc box in which was a small opening covered with a film of
aluminium, thin enough to allow the rays to pass through. The end of this wire was
placed in front of the window and close to it, with the electroscope clear of the direct
path of the rays. The box itself was connected to earth and set in position, with its
window opposite that of the discharge tube.
Using this apparatus, Lenabd found that positive and negative charges alike were
completely dissipated by a single discharge through the tube when the aluminium
windows were at any distance up to 4 centims. apart. At greater distances than
this a similar but only partial discharging of both kinds of electricity occurred when
the same amount of rays was used.
This loss of charge was no doubt brought about by means of the ionization in the
air surrounding the conductor. The known behaviour of an ionized gas, however,
would have led one to expect a somewhat different result, especially in regard to the
effect obtained with short distances between the windows. When an insulated metal
conductor is placed in air ionized by Rontgen rays, Zeleny* has shown that, owing
to the greater velocity with which the negative ions diflRise, this conductor takes up a
small negative charge, while the gas itself is left with a positive one. If then the
ionizations in the two cases are of the same nature, one would have expected that in
Lenard's experiments the wire and electroscope would not, under any circumstances,
have been finally discharged completely, but would have been left with at least a small
negative charge. When, further, it is remembered that the impinging cathode rays
themselves carried a negative charge to the wue, this fact affords an additional reason
for expecting such a result.
Fi^.H.
Now the goldleaf electroscope, as used by Lenard (Exner's type), was not
sensitive to small differences of potential, and it was consequently not a suitable
instrument for the detection and measurement of effects of this kind. As the
explanation of his results seemed, then, to be connected with this lack of sensibility
in the measuring instrument, his experiments were repeated, and a quadrant elec
trometer was used in place of the electroscope.
♦ 'Phil. Mag.,' July, 1S98, p. 13^.
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54 MR J. c. Mclennan on electrical conductivity in gases
The arrangement was that shown in fig. 2. A copper wire, terminated by a disc, A,
of the same metal, was insulated by ebonite fix)m an earthconnected copper tube B,
through which it passed to the electrometer. To this tube there was fastened, as
shown in the figure, a large, finelymeshed copper gauze which completely protected
the disc from electrostatic induction. The tube B also carried a short concentric
cylinder a, made of copper, which could be slid out when desired so as to surround
the projecting end of the wire and the disc.
On placing this apparatus in front of the aluminium window so that the cathode
rays fell on the disc, it was found that, although the rays caused a discharging of
positive and of negative electricity, still in no case observed was a negative chai'ge
on the disc and wire ever completely dissipated.
Negative charges fell, however, to limiting values, represented in some cases by
potentials of the order of '25 volt, and then remained stationary. In the case of
initial positive charges the discharging was not only complete but the disc also gained
this limiting negative charge. A similar charging action was observed when there
was no initial charge on the disc.
Here the disc was subjected to two influences, namely, the cathode rays canying
a negative charge to it and the ionized gas about it acting as a conductor and tending
to discharge it. This limiting charge can, then, just as in the case already cited, be
looked upon as representing a state of equilibrium in which the convection to the
disc was just equal to the conduction away fix)m it.
As the electric field produced by a given charge on the disc would vary with
the distance between it and neighbouring conductors at a different potential, the
conduction from the wire could consequently be increased or decreased according
as an earthconnected conductor was brought close to the disc or removed farther
from it. If then a means were devised of altering in this way the conduction without
altering the intensity of the rays impinging on the disc, the value of this limiting
charge could be subjected to definite variations.
The sliding cylinder a affoixled a simple means of accomplishing this result. If
when the tube was excited a stationary state was reached, with this cylinder shoved
well back, and it was then brought forward over the wire and disc, the limiting
negative charge at once dropped and assumed a steady but smaller value. In order
to restore the charge to its original value it sufficed merely to slide the cylinder back
to its former position.
Another simple verification of this view was afforded by the use of a blast of air.
If when the rays were impinging on the disc a blast of air was directed towards it
and at right angles to the rays, the limiting charge at once increased to another
limiting valut^, and when the blast stopped it again dropped to its original amount.
As the velocity of the cathode rays has been estimated by J. J. Thomson* to be of
the order of 10^^ centims. per second, it is clear that any ordinary bliist could produce
♦ * Phil. Mag.,* October, 1897, p. 315.
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TRAVERSED BY CATHODE RAYS. 55
very little effect on the motion of these carriers. On the other hand, the velocity of
the ions in Rontgenised air has been found by Rutherford* to be about 1 '6 centims.
per second, under a field of a volt a centimetre, and consequently of the order of that
of the blast. In the experiment described, the effect of the blast, therefore, was to
decrease the conduction away from the electrode by removing the ionized gas ; and as
no change was made in the intensity of the rays impinging on the disc, this con
sequently produced an increase in the residual charge. This increase, however, did
not go on indefinitely, but ceased when the field it set up was sufficient to neutralise
the effect of the blast ; hence the second stationary value for the charge.
Another means of increasing this limiting charge was afforded by the removal of
the air surrounding the electrode. To show this the gauze cap was removed from
the apparatus in fig. 2, and the metal tube siurounding the wire was brought forward
and sealed to the anode of the discharge tube. The arrangement is shown in fig. 3,
With this apparatus it was found that, as the exhaustion proceeded in the chamber
B, the negative charge received by the electrode A gradually increased, until finally,
at a very high vacuum, a momentary discharge of the rays was sufficient to raise its
potential beyond the range of the electrometer. This result, therefore, confirms the
explanation already given of the discharging action of the rays. In a recent paper
by LENARDt this charging action of the cathode rays in a high vacuum was described,
but its connection with the ionized air surrounding the electrode was not brought
out. From the experiments just described it is clear that, while this action is directly
due to the fact that the cathode rays carry a negative charge, the extent of the effect
obtained in all cases depends to a very great degree upon the opposing influence
exerted by the ionized air surrounding the electrode upon which the rays fall.
4. Ionization not due to Rontgen Rays.
It has been thought by some that the ionization pixxluced by c«.thode rays was due
to Rontgen rays, which might possibly be sent out from the window at the same
time. The results of experiment are, however, entirely opposed to this view.
* *Phil. Mftg.,' Noveml>er, 1897, p. 436. t * Wied. Ann.,' vol. 63, p. 253.
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56 MR. J. c. Mclennan on electrical conductivity in gases
In order to investigate the point an apparatus similar to that shown in fig. 3
was adopted. Different thicknesses of aluminium foil were in turn used for the
window, and the air in the chamber B was kept at a pressure low enough to
absorb but little of any radiation coming from the window, and yet sufficiently
high to aflford considerable conductivity when ionized.
With diflferent thicknesses of the foil down to "04 millim., it was found that the
electrode A did not gain any charge when the tube was excited. Further, if in
these cases a charge, either positive or negative, was given independently to the
electrode, this charge was maintained when the discharge passed in the tube, and
no leak occurred. But when the window was made of foil '008 millim. in thickness,
the effect obtained was such as that already described in the last paragraph.
Under these conditions the electrode A, if carrying initially a positive or a negative
charge, finally assumed a stationary state, in which it carried a definite negative
charge whose value, as has already been pointed out, depended upon the pressure
of the air in the chamber B. As, then, no leak from the electrode occurred when
the aluminium was '04 millim. in thickness, it seems justifiable to conclude that if any
Rontgen rays were present under these circumstances they were of an extremely
weak character. If Rontgen rays of even very moderate intensity had entered the
chamber, a leak would have taken place which could have been observed. In practice
the aluminium foil used in my experiments was about '008 millim. in thickness, and
with this foil intense ionization was observed. From the known character of Rontgen
rays, it was quite impossible for this great ionization to be produced by rays which
could be absorbed by a layer of aluminium "032 millim. — ^the difference in thickness
of the two windows.
Again, an ordinary focus tube illustrates very well the fact that the Rontgen
rays produced issue in a large measure firom the face of the anticathode, upon
which the cathode rays fall, while the radiation appearing to come fi:'om the opposite
face is always very weak. The theory now generally accepted is that the Rontgen
rays are electromagnetic pulses sent through the ether when the moving electrified
particles which constitute the cathode rays are suddenly stopped. If then the
Rontgen radiation sent out in the direction of propagation of the cathode rays, when
these carriers were stopped by foil '04 millim. in thickness, was at most but very
feeble, it appears highly improbable that a strong radiation of this kind could be
produced by those carriers that passed through the thinner foil without being
stopped.
The conductivity produced in a gas by cathode rays is, moreover, far in excess of
that excited by even the strongest Rontgen rays. In order to make a direct com
parison, measurements were taken of the ionizations produced in the same chamber
by both radiations, and the following illustration gives an indication of their
respective efficiencies. By using the apparatus shown in fig. 1, it was found that,
under the action of cathode rays with a saturating intensity of field, a capacity of
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THAVEBSED BY CATHODE RAYS. 57
750 electrostatic units attached to the electrode B gained in 15 seconds a charge
represented by 300 divisions on an arbitrary scale. A Rontgen ray focus tube
giving out very strong rays was then used in place of that for producing the
cathode rays, and was excited by an induction coil capable of giving a 50centims.
spark. Under these circumstances, with the same field, which was also in this case
a saturating one, a capacity of 150 electrostatic units was charged in one minute to
an amount represented by 20 on the same scale. This case, which is an extreme
one, shows that the ionization by cathode rays was about 300 times that due to an
intense Rontgen radiation. In the present investigation these latter rays, even if
they did accompany the cathode rays, must have been very feeble, and could there
fore only exert an ionizing influence which may be left out of consideration.
The known action of a magnetic field naturally suggested itself as a means of
sifting out the cathode from any accompanying Rontgen rays. The intensity of the
cathode rays, however, soon falls off owing to their rapid absorption by the air, and
on this account it was necessary to place the chamber in which the ionization was
measured close up to the discharge tube. Under these conditions it was found
impossible to deflect the rays outside the tube without also deflecting those inside.
This difficulty consequently rendered the test indecisive, and the method had to be
abandoned.
5. Discussion of Methods for Measuring the Ionizations produced in Different Gases.
In the construction of Rontgenray bulbs, the disengagement of gas from the
electrodes and the inside of the glass is facilitated by the application of heat to the
tube. In the case of Lenard tubes, however, the joints are made of wax, and the
final stage of exhaustion cannot be hastened by adopting this device. In practice a
tube was kept attached to the mercury pump, and exhausted while the discharge
was passing through it. After some hours of this procedure the coil was stopped,
and the exhaustion was continued until only some traces of air were being taken
over. On then exciting the tube, the vacuum was found to be sufficiently high for
the cathode rays produced to penetrate the aluminium window. After running the
coil for a short time, a small quantity of gas acciunulated in the tube, and the
pressure rose so high that the rays ceased to be propagated outside. After this air
had been removed the vacuum again became good, and the original intensity of tlie
rays was restored. As the ionizing power of the rays was very great, charges
sufficiently large to be accurately measured were easily accumulated by exciting the
tube only for short periods. By following this course quite satisfactory results
were obtained and much loss of time was avoided.
On account of this running down of the discharge tube, it was impossible, in
comparing the ionizations in two different gases, to use an apparatus with a single
chamber, such as that shown in fig. 1. In order to obtain accurate results, it was
VOL. CXCV. — A. I
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58 MR. J. c. McLennan on electrical conductivity in gases
necessary either to have a constant source of rays, or else to be able to ascertain the
relative intensities of the rays used with the different gases.
One method which suggested itself was the use in series of two chambers, such as
that shown in fig. 1. By inserting a thin aluminium membrane between them a
different gas could be put in each chamber, and a single pencil of rays could be used
to produce the ionization in both chambers. With this arrangement it was thought
that the ionization obtained in the first chamber might perhaps bear a constant
ratio to that produced in the second. But this relation was not found to hold, and
further, as the cathode rays are rapidly absorbed, the amount of ionization obtained
in the first chamber was so very much greater than that in the second, that even
if the ratio had been fairly constant the method would not have been at all
satisfactory.
This led to a trial of two receivers in parallel. Although the cathode rays on
issuing from the window diverge very greatly, mechanical difficulties made it im
practicable to receive part of the issuing rays in each chamber, and so recourse was
had to the use of two windows. With a single large disc as cathode, a stream of
rays was received in each of the chambers. The ratio of their intensities, however,
as measured by the ionizations they produced, did not remain constant but varied
quite irregularly. The explanation of this is probably found in a paper by A. A. C.
SwiNTON,* where he points out that the carriers are shot off in a hollow cone from
the cathode, and that the dimensions of such a cone of rays vary with the degree oi
exhaustion in the tube. Besides, the aluminium windows were opposite to eccentric
points on the cathode, and the ratio of the intensities of the two pencils was in this
way greatly influenced by slight variations in the directions of the rays within the
tube. A cathode formed of two small discs was then tried, and the results obtained
were very satisfactory. The ratio of the discharges from the windows was in this
case quite constant, and it was therefore possible to make measurements with con
fidence. The main difficulty of the investigation was in this way overcome, and the
method was applied to obtain among other things a knowledge of — 1, the absorption
of the rays ; 2, the ionizations produced by them in air at different pressures ; and
3, the relative ionizations in different gases.
6. Description of Apparatus used.
A diagram of the apparatus is shown in fig. 4, and the way in which the connec
tions were made is exhibited in fig. 5. The exciting tube was slightly over
3 centims. in diameter. The two discs of the cathode were each about a centimetre
in diameter, and they were placed with their centres directly in front of the
alimiinium windows. That portion of the apparatus in which the ionizations were
* *Proc. Roy. Soc.,' vol. 61, p. 79 (1897).
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TRAVERSED BY CATHODE RAYS. 59
measured consisted of two chambers, A and B, each made of brass and similar in
form to that shown in fig. 1. The two electrodes C and D were held in position by
ebonite plugs, which closed the ends of the receivers and at the same time served as
insulators.
el*^
$arf^
In each experiment the receivers themselves were well earthed, and also, initially,
the electrodes C and D. As the electrostatic induction was very intense in the
neighbourhood of the discharge tube, it was found necessary to take special
precautions in regard to the earth connections. Wires of but very small resist
ance were used, and these were led to water mains and all the joints carefully
soldered.
The two chambers were separated by a disc of ebonite, and to its faces were
attached thin brass plates, a and h. By means of wires passing out through the
I 2
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MR. J. c. Mclennan on electrical conductiyity in gases
ebonite to the battery of storage cells E, these plates could be charged to any
desired potential, high or low. As the electrodes and the walls of the receiver were
earthed, this afforded a means of setting up in each chamber a field which could be
readily modified. The fields themselves, moreover, were quite distinct, each disc
serving as a screen to cut off any action arising fi'om the other.
Each of the chambers was provided with a projecting shoulder, which slid over a
corresponding one on the anode surrounding the window opposite. By coating these
joints with wax the chambers were then not only made airtight, but also were
entirely separated from each other.
In the apparatus used, the diameter of the chambers A and B was about 3 centims.,
and the distance between each of the electrodes and its corresponding plate a or
h about 1'6 centims. The diameter of the narrow cylinders which admitted the rays
to the chambers was 3 millims., and the distance between the aluminium windows
and points corresponding to the centres of the electric fields was about 2 centims.
\>XNXX\XXX\XX\\\\\\\\\\\\N>X\\X\
Each of the electrodes was connected to an air condenser, whose capacity was
about 600 electrostatic units. These condensers, G and H, were each made of two
sets of parallel plates separated by small ebonite supports. The plates were made by
coating both .sides of a sheet of glass by a single sheet of tinfoil. In this way plates
tolerably plane were obtained, and yet difficulties arising from electric absorption
were avoided, the glass merely serving as a support for the foil plates.
The measurements were made with a quadrant electrometer F, and the tube was
excited by a 50centims. sparklength induction coil, whose positive terminal, together
with the anode of the tube, was kept to earth. This coil was provided with an
Apps interruptor, and besides being very powerful was also very efficient. It
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TRAVEESED BY CATHODE RAYS. 61
required a potential of only eight volts to excite it, and with the interruptor working
slowly, this was suflScient to produce sparks of the maximum length in air at normal
pressure. In practice, the interruptions were made at the rate of 20 to 25 per
second.
7. Explanation of the Method adopted foi* Comparing Ionizations.
It is well known that, in conduction in Rontgenised gases, and in gases acted upon
by uranium radiation, the current of electricity obtained does not increase in propor
tion to the electromotive force applied. The current, after reaching a certain critical
value, becomes practically stationary, and increases but very little when very large
increases are made in the electromotive forces. This maximiun, or saturation current,
was also found to characterise the conductivity produced by the passage of cathode
rays through a gas. With Rontgen or uranium radiation, a field of 400 or 500 volts
a centimetre has been found to give saturation in most simple gases; but with
cathode rays it was necessary to apply fields of much stronger intensity.
As already stated, the distance between either of the electrodes C and D, fig. 5,
and the dividing partition was about 1 '6 centims. In order to ascertain the saturating
electromotive force, the plate h was kept at a very high potential, while that of
a was gradually increased fi'om zero. At each stage the ratio of the currents
obtained in the two chambers was noted, and it was not until a potential of about
900 volts was applied to a that an approximation to the saturation current was
obtained in the chamber A. With a potential difference of 1200 volts the increase
in the current was small, and an increase only slightly larger was obtained with a
potential of 1600 volts, or 1000 volts a centimetre. This small increment in the
current very probably arose from the influence of the field itself. It may be that in
certain parts of the receiver the rays, acting in conjunction with the applied differ
ence of potential, had not quite sufl&cient intensity to produce dissociation. An
increase in the field under these circumstances would produce greater ionization, and
consequently a larger current would be obtained. As this field of 1000 volts a centi
metre practically produced saturation currents in both chambers, it was used through
out in measuring the ionizations. Sparking was prevented by using in the charging
circuit liquid resistances, such as xylol.
An explanation of the saturation current is that the number of ions used up by the
current in a given time is exactly equal to the number .produced by the rays in the
same time, or in other words, the ions are removed so rapidly by the applied field
that recombination is practically eliminated. The saturation current is then a direct
measure of the ionization produced, and in order to compare the ionizations in any
two gases, it suffices to measure their saturation currents. In this investigation the
saturating electromotive force was applied to the plates a and 6, the discharge
tube was then excited, and the currents obtained were used to charge up the con
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62 MR J. C. McLENNAN ON ELECTRICAL CONDUCTIVITY IN GASES
densers G and H. The discharge having been stopped, the potentials of the two
condensers were then successively determined.
As the effective capacity of the electrometer was the same fraction of that of each
of the two equal condensers, the deflection readings were direct measures of the
charges obtained. The charging of both condensers proceeded for the same time,
and consequently the electrometer deflections were also direct measures of the
saturation currents, and therefore of the ionizations in the two chambers.
The method possessed the advantage of being independent of the time of charging
and of the strengths of the rays coming from the two windows, provided only that
the ratio of their intensities remained constant. In using the electrometer the
needle was kept at a high potential, and one pair of quadrants always connected to
earth. Though with this arrangement slow losses from the needle occurred, yet the
short interval required for the two readings made the gradual change in the effective
capacity of the electrometer inappreciable.
In practice, the electrometer was initially connected to one of the condensers, and
the tube allowed to run until a suitable deflection was obtained. After noting this
reading, the electrometer, having been put to earth, was then connected to the other
condenser and the second reading taken. In this way the ratio of the ionizations in
the two chambers was obtained.
From the experiments described in Section 2, it is clear that the signs of the
charges obtained in the condensers depended on the signs of the charges given to
a and h by the battery. In case these plates were positively charged, the
charges collected were positive, and were due entirely to ionization. With a nega
tive field, however, the negative charges obtained included not only negative ions
produced by the rays, but also the negative carriers, constituting the rays, that were
stopped in their motion by the gas. For this reason the positive field was always
used, and consequently the charges obtained gave a measure of the number of ions
produced in the gas by the passage of the rays.
8. Ionization in different Gases at the Same Pressure.
To compare the ionization in a selected gas with that in air at the same pressure,
the saturating electromotive force was applied to the plates a and &, fig. 5. The
two chambers A and B were first filled with air at atmospheric pressure, and a series
of readings taken, the mean of which gave the ratio of the saturation currents in
the two chambers. The air was then removed from A, and the gas to be tested
introduced. A set of readings similarly taken gave a ratio for the saturation
current obtained with the given gas in A, compared with that obtained with air in
B. The combination of these results gave the ratio of the saturation current in A,
when filled with the given gas, to that in the same chamber when filled with air.
This ratio was, consequently, the ratio of the ionization produced in the selected gas
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TRAVERSED BY CATHODE RAYS.
to that produced in air at an equal pressure under the action of cathode rays
entering the chamber with the same intensity in both cases.
The results obtained from this method for hydrogen, air, and carbon dioxide are
given in the first column of Table I. In the second column are given the relative
ionizations found by J. J. Thomson* for these same gases when ionized by Rontgen
rays of constant intensity.
Table I.
Name of gas.
Column I.
Column 11.
Ionization by
cathode rays.
Ionization by
Rontgen rays.
Hydrogen
Air
Carbon dioxide . . .
265
100
•34
•33
100
140
These numbers, it will be seen, present a very marked difference. In the one case
the ionization decreased as the density of the gas traversed increased, while in the
other a law directly the reverse of this was followed.
One explanation of this difference in the results is that the character of the
ionization under cathode rays may be essentially different from that produced by
Ilontgen rays. Apart from these numbers, however, there seems to be but little
ground for this view. Strong experimental evidence now exists to support the
assumption that the cathode rays consist of small particles of matter carrying nega
tive charges of electricity. We may therefore regard the ionization they produce as
being due to their impinging on the molecules of a gas, and to the consequent
breaking up of the latter. On this hypothesis it is not clear that the resulting ions
should differ in character from those produced under the influence of Rontgen
radiation.
It appeared rather that the true explanation was to be found in the varying
absorbing powers of the different gases. LENARD,t who studied these rays by the
fluorescence they excited, found that the absorption of cathode rays by gases at
atmospheric pressure was considerable. He was also led by his experiments to
propound the law, that while different gases at the same pressure absorbed the rays
to different degrees, yet their absorption depended only upon the densities of the
gases, and not upon their chemical composition.
In the apparatus here used, the distance traversed by the rays after they left the
discharge tube until they reached the centre of the field where the ionization was
♦ *Proc. Camb. Phil. Soc.,* vol. 10, Part I., p. 12,
t • Wied. Ann.,* vol. 56, p. 265 (1895).
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64
MR. J. c. Mclennan on electrical conductivity in gases
measured, was about 2 centims. From Lenard's conclusions, it is obvious that in this
distance the absorption of the rays by carbon dioxide would be greater than by air,
and very much greater than by hydrogen. The effective intensities of the rays in the
three gases at the same pressure would then be very diflferent, and numbers such as
those given in Column I. follow naturally under these circumstances, without assuming
any difierence in the character of the two ionizations.
9. Ionization in Air at Different Pressures.
In order to study more closely the influence of absorption, a number of experiments
were carried out similar to that just described. The same apparatus was used, and the
same method followed, but the ionizations, instead of being measured in different
gases at the same pressure, were determined for the same gas at different pressures.
Table IL
Pressure.
Ionization measured.
millims.
767
100
530
144
340
192
205
232
104
268
53
274
Between 40 and 45 millims. a sudden large
increase was obtained in the ionization.
This was found to be due to the action
of the field itself in dissociating the gas.
The results obtained with air are shown in Table II. The pressures are expressed
in heights of columns of mercury at the same temperature. The ionizations given
are relative, that corresponding to atmospheric pressure being taken as unity, and
each value is the average of a large number of readings.
The results are also shown graphically in fig. 6, where the abscissae represent
pressures, and the ordinates corresponding relative ionizations.
The numbers show that as the pressure decreased the ionization obtained with a
saturating electromotive force steadily increased, until a pressure of about 75 millims.
of mercury was reached. This result, though surprising, can be readily explained
by the great absorption of the rays at atmospheric pressure.
The rays had to travel at least 1 "5 centims* from the window before they reached
that part of the chamber from which the saturation current was obtained. For this
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TRAVEliSED BY CATHODE RAYS.
65
reason their efTective intensity was very largely determined by the pressure of the
gas traversed.
While a diminution in the pressure would not affect the original intensity of a
pencil of rays issuing from the window, it would, owing to a decrease in the absorp
tion, increase the ionizing power of this pencil at the centre of the receiver. In this
way, although the available amount of matter to be ionized was lessened by lowering
the pressure, it could happen that the resultant ionization, as measured by the satu
ration current, would at first exhibit increasing values. This in all probability
accounts for the numbers obtained in Table H.
Now frora this point of view such a condition would only hold down to a stage
when the two influences produced equal effects. The ionization would then be a
maximum, and would afterwards fall off with diminishing pressures. Although the
numbers obtained for the saturation current do not show definitely that a maximum
value was obtained for the ionization, still there are indications from them, as the
curve shown on fig. 6 illustrates, that the maximum value was reached at a pressure
of about 75 millims. of mercury.
/on/zat/ofTS
4
3
^^
.i^
2
/
^.^
^
\
lOO ?(
90
fi
00
5
10
m
H>
—
70
"T
55"
A
W i
}
PRsaauRcs IN MMS.
Fig. VI.
As indicated in Table II., the conditions of the experiment made it impossible
to measure the ionization in air at pressures much below 50 millims. At about
40 millims. pressure a sudden large increase was obtained in the value of the satura
tion currrent, whch was found to be due to the influence exerted by the applied field
in breaking down the gas. At these low pressures the electric intensity, which was
1000 volts a centimetre, wassufiicient to dissociate the attenuated gas and to produce
a discharge on its own account between the electrodes. This was shown by simply
connecting the electrometer to one of the electrodes, C for example, and applying the
potential difference without exciting the discharge tube. On then exhausting the
VOL. CXCV. — A.
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66 MR. J. c. McLennan on electrical conductivity in gases
chamber, the electrometer showed no leak until the critical pressure was reached,
when it inmiediately began to charge up.
10. Ionization in a Gas independent of its Chemical Composition.
An important result in connection with these experiments is the agreement
exhibited between the number given in Table I. for the ionization in hydrogen at
atmospheric pressure and that given in Table II. for the ionization in air at a
pressure of 53 miUims.
Here two gases, hydrogen and air, were introduced in succession into the same
measuring chamber and adjusted to the same density. Cathode rays of the same
intensity were projected into this chamber in the two cases, and these rays, after
traversing a certain length of the gas, reached a point where the ionization they
there produced was measured. The values obtained show that under the circum
stances the same number of ions was produced in both gases.
Since the rays issuing from the window were in both cases of the same intensity, it
follows from Lenard's absorption law that the disposition of the rays, their actual
intensities, and the quantities of them absorbed from point to point in the chamber,
were precisely the same in both gases. Under these circumstances, therefore, the
equal ionizations obtained in hydrogen and in air at the same density not only form
a confirmation of Lenard's absorption law, but also show that where equal absorption
occurs equal ionization is produced.
In the case of Rontgen radiation, Rutherford* has made a determination of the
relative absorbing powers of a number of gases. Taking I to denote the intensity
of the rays on entering a particular gas, and le"^ their intensity after traversing a
length X, he has found that the values of the coefficient of absorption for the different
gases practically represented the relative conductivities produced in these same gases
by Rontgen rays. It is thus interesting to note that with cathode rays, just as with
Rontgen rays, equal absorption gives equal ionization.
To test still further the accuracy of this conclusion a detailed examination was
made of the ionization produced in a number of different gases. Throughout the
experiments air in the chamber B, fig. 5, was taken as the standard. In some
comparisons this air was kept at atmospheric pressure, while in others lower
pressures were taken, the pressure selected being maintained through each complete
determination. In making a comparison the chamber A was fiJled in turn with the two
gases to be examined, and their pressures were adjusted so as to reduce them to the
same density. Two ratios were in this way found for the ionizations in the chambers
A and B, and as the influence of absorption was eliminated on account of the equal
densities, these ratios represented the relative ionizations in the two gases under
cathode rays of the same intensity.
* * Phil. Mag.,' April, 1897, p. 254.
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TRAVERSED BY CATHODE RAYS.
67
These ratios were determined by taking the mean of a number of readings.
Samples of the results obtained in five different comparisons are given in Tables III.,
IV., v., VL, and VIL, the numbers being consecutive readings with each gas in A.
They represent very well the working of the method. Although the variations were
considerable, similar ones occurred in both sets of observations in each comparison, and
as the number of readings taken was very large, any errors were in a great measure
compensated.
Table III. — Oxygen and Air.
Air in both chambers at
Oxygen in A at 6751 millims.
7467 millims.
Air in B at 7467 millims.
Ionization in A.
Ionization in B.
Ionization in A.
Ionization in B.
109
100
140
100
137
128
99
154
139
9
124
120
»
135
154
1
125
141
}
107
120
9
141
126
9
154
141
9
131
133
9
132
100
134
100
Table IV. — Nitrogen and Air.
Air in A at 7343 millims.
Nitrogen in A at 757 millims.
Air in B at 757 millims.
Air in B at 757 millims.
Ionization in A.
Ionization in B.
Ionization in A.
Ionization in B.
110
100
104
100
102
99
M2
121
J
J
103
105
J
9
134
129
i
103
110
9
106
118
9
M5
104
9
M2
107
9
108
100
n
106
Ml
100
MO
100
K 2
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68
MR. J. C. McLENNAN ON ELECTRICAL CONDUCTIVITY IN GASES
Table V. — Carbon Dioxide and Air.
Air in both chambers at
7727 millims.
Carbon dioxide in A at
5047 millims.
Air in B at 7727 millims.
Ionization in A.
Ionization in B.
Ionization in A.
Ionization in B.
122
100
117
100
112
))
116
»
117
)
123
9)
13S
}
131
))
102
»
137
})
130
1
100
yj
111
y
121
))
117
y
124
))
103
f
131
yy
123
>
100
i»
117
1
00
120
100
Table VI.— Hydrogen and Air.
Air in both chambers at
Hydrogen in A at 770*9 millims.
53*2 millims.
Air in B at 53*2.
Ionization in A.
Ionization in B.
Ionization in A.
Ionization in B.
158
100
152
100
177
182
}}
164
191
)
141
163
y
162
158
y
163
180
y
179
170
y
173
132
y
185
175
y
181
204
>
168
100
171
100
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TRAVERSED BY CATHODE RAYS.
69
Table VII. — Nitrous Oxide and Air.
Air in both chambers at
759 minima.
Nitrous oxide in A at
499 millims.
Air in B at 759 millims.
Ionization in A.
Ionization in B.
Ionization in A.
Ionization in B.
108
100
103
100
MO
121
99
124
M5
9)
112
107
))
•99
113
11
108
107 „ 1
111
117
99
123
102
99
107
105
))
112
110
»>
111
100
110
100
Care was taken to insure the purity of the gases, and they were also well dried
before being passed into the ionizing chamber.
The oxygen was prepared electroly tically, and was freed from ozone by being passed
through a strong solution of potassium iodide and caustic potash.
The nitrogen was prepared by gently heating a mixture of ammonium chloride with
a nearly saturated solution of sodium nitrite. The gas given off was passed through
a Utube containing strong caustic potash, and also through a second containing
concentrated sulphuric acid. A Kipp apparatus was used for the preparation of carbon
dioxide, which was made in the ordinary manner by allowing dilute hydrochloric acid
to act on marble. In making hydrogen a Kipp apparatus was also used, dilute
sulphuric acid being allowed to act on zinc. The gas was passed through a strong
potassium permanganate solution, and then through a Utube containing a strong
solution of caustic potash.
The nitrous oxide was prepared by heating ammonium nitrate in a flask, and the
gas was collected over water.
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70
MR. J. c. McLennan on electrical conductivity in gases
Table VIII. — Summary of Measurements.
Gases compared.
Pressures. Ionizations.
1
Air mean of 30 readings
Oxygen „ 30 „
millim*.
7467
6751
131
132
Air mean of 25 readings
Nitrogen .... » 25 „
7343
757
Ml
109
Air mean of 30 readings
Carbon dioxide . . « 30 „
7727
5054
120
118
Air mean of 18 readings
Hydrogen .... „ 18 „
532
7709
170
179
Air mean of 23 readings
Nitrous oxide . . i, 23 „
759
4993
109
110
A summary of complete sets of observations on the different gsises is given in
Table VIII. This statement includes the number of readings made in each case and
the pressures at which these were taken. The ionizations quoted are the averages of
the several sets of readings.
The close agreement exhibited by the numbers corresponding to each comparison
fully bears out the conclusion deduced from the earlier experiments. It not only
forms a striking corroboration of Lenard's absorption law, but also shows that the
ionization follows an analogous one, which may be stated thus : — When cathode rays
of a given strength pass through a gas, the number of ions produced per second in
1 cub. centim. depends only upon the density of the gas, and is independent of its
chemical composition.
The similarity in the laws of absorption and ionization, holding, as it does, with so
many gases over such a wide range of pressures, is a clear indication that when
cathode rays are absorbed to a certain extent, the positive and negative ions produced
by these absorbed rays are of a definite amount, which bears a constant ratio to the
quantity of the rays absorbed ; that is to say, the absorption of a definite amount of
radiant energy is always accompanied by the appearance of a fixed amount of potential
energy in the form of free ions.
This granted, it follows that in order to ascertain the relative ionizations pro
duced in any two gases by cathode rays of the same intensity, it is sufficient to
determine the absorbing powers of the two gases for the same rays. In other words,
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TRAVERSED BY CATHODE RAYS. 71
the coefficients of ionization for a series of gases are fully determined when the
coefficients of absorption for these same gases are known.
The existence of this general relation between absorption and ionization for both
cathode and Rontgen rays is especially interesting when we remember that the two
radiations are so very different in many respects.
In the one case, according to the generallyaccepted view, the rays consist of
small charged particles of matter moving with high velocities in space, while in the
other they are supposed to consist of electromagnetic impulses propagated in the
ether. With the one the dissociation is in all probability brought about by a series of
impacts between the moving particles and the molecules of the gas ; with the other
it seems to be due to the direct action of the intense electric field forming the impulse.
Again, while the absorption of cathode rays depends only upon the density of the
medium traversed, the absorption of Rontgen rays, according to Rutherford's
results, does not seem to depend to any great extent upon the molecular weight of
the gas. But while all these differences exist in the two radiations, with both of
them it holds good that the same number of ions are always produced in a gas when
the same amount of rays traversing it are absorbed.
11. Comparison of Ionizations produced hy Cathode and hy Rontgen Rays.
The method just described gives definite and conclusive information regarding the
ionizations produced by cathode rays in gases of the same density ; but where the
gases are of different densities, it cannot be satisfactorily applied. As stated in
Section IX., the rays, after entering the ionizing chamber, must travel some distance
before reaching that part of the field fi'om which the current is drawn. On this
account, though rays entering the chamber may originally be of the same strength,
still their effective intensities become at ordinary pressures quite different, when the
gases traversed are not of the same density.
Also as it is impossible to define exactly the disposition of the electric field within
the chamber, these effective intensities cannot be calculated with any degree of
accuracy.
A difficulty arises, too, from the dispersion of the rays. As shown by Lexard,
they issue from the window in a pencil whose form is greatly influenced by the
density of the gas traversed. At very low pressures they pass through the
aluminium window practically without deviation, but as the pressure increases, they
spread out until finally they issue in all directions.
The conclusion arrived at in the last section, however, suggests a means of calcu
lating the ionization which would be produced by rays of constant intensity in
different gases at the same pressure.
Lenard,* who investigated the absorption powers of a number of gases at different
* * Wied. Ann.,' vol. 56, p. 258.
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72
MR. J. c. mclen:nan on electrical conductivity in gases
pressures, has shown that for any particular gas the coefficient of absorption varies
directly as the pressure. In the case of air, taking I to denote the intensity of the
rays issuing from the window of the discharge tube, and le"^ their intensity at a
distance x from the window, he found for \ the values given in Table IX.
Table IX.
Air pressure
•
Coefficient of absorption.
millims.
760
343
331
151
165
•661
837
•396
405
•235
193
•117
100
•0400
27
•0166
•78
•00416
These numbers, it wiU be seen, amply support Lenard's conclusion. Similar
tables, given by him for a number of gases, all exhibit the same relation between the
values of X and the corresponding pressures of the gas.
Now, if the values of the coefficient of absorption are taken to represent the rela
tive ionizations produced in a gas, at a point where the pressure is varied but the
intensity of the rays kept constant, it follows from Lenard's numbers that the
ionization in any particular gas would vary directly as the pressure to which it was
subjected.
This result, which follows as a deduction from the preceding experiments, has also
been found experimentally by Perrin* to characterise the ionization produced by
Rontgen rays. It is true that with Rontgen rays a number of experimenters have
found quite different relations to hold between the ionization and the pressure ; but
in most cases they have vitiated their results either through omitting to use satu
rating electromotive forces, or through neglecting to arrange their experiments so as
to eliminate the metal effect observed by Perrin.
With uranium radiation also, RuTHERFORDt has found the ionization to be propor
tional to the pressure of the gas traversed.
The direct experimental verification of a law of this kind is always accompanied by
a serious difficulty. The law has reference to the action of rays whose intensity is
constant throughout the region ionized. With rays that are easily absorbed by
gases at ordinary pressures, this condition can be realised either by the use of very
thin layers of gas or by investigating the ionizations at very] low pressures. Owing
* *Comptes Rendus,' vol, 123, p. 878.
t *Phil. Mag.,' January, 1899, p. 136.
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TRAVERSED BY CATHODE RAYS. 73
to mechanical difficulties, however, the former method is generally impracticable,
while the action of the applied electric field in breaking down the insulation of the
gas precludes the use of the latter artifice.
It is then open to measure the ionizations produced by rays, traversing layers of
gas of considerable thickness. But before any relation connecting ionizations and
pressures can be deduced from such measurements, it is necessary to have definite
information regarding the absorptive powers of the gases at different pressures, and
to know exactly the form and dimensions of the region from which the ions are
drawn.
Although the absorption laws for cathode rays have been fully developed by
Lenard, and are quite definite and clear, it is scarcely possible to define even
approximately the region in the ionizing chambers (fig. 5) from which the ions go to
make up the saturation current.
On this account a direct verification of the proportionality law is not possible ;
but, as already pointed out, the results of the experiments described in Section X.
strongly support the conclusion that, in the case of a gas subjected to increasing
pressure, the ionizations produced by rays of constant intensity bear the same ratio
to each other as the coefficients of absorption corresponding to these pressures.
If, then, the ionization in a gas varies with the pressure, it follows at once that if
rays of the same intensity were allowed to traverse thin layers of different gases at a
constant pressure, the ionizations produced would be directly proportional to the
densities of these gases.
Take, for example, carbon dioxide and air. It has been shown that the ionization
produced in carbon dioxide at a pressure of 5047 millims. of mercury is the same
as that produced in air at 7727 millims. by rays of the same intensity.
According to the proportion law the ionization produced by these same rays in
CO2 at 772 7 would then be just r53 times that obtained at the lower pressure ;
that is, with rays of the same intensity the ionizations in carbon dioxide and in air
would be to each other as 1*53 to 1 when these gases were subjected to the same
pressure.
A similar conclusion may be deduced from a consideration of the other gases
examined. Hence, on this view, the relative ionizations produced by rays of
constant intensity in a series of gases subjected to the same pressure would be
expressed by the numbers which under these circumstances give their relative
densities.
These numbers are given for the gases examined in Column I., Table X., while in
Column II. are given the values found by J. J. Thomson* for the relative ionizations
produced by Rontgen rays of constant intensity in the same gases.
* *Proc. Camb. Phil. Soc.,* vol. 10, Part I., p. 12.
VOL. CXCV. — A. L
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74
MR. J. C, McLENNAU on electrical CONDUCTIVITY IN GASES
Table X.
Column I.
' Column XL
Gases examined.
Densities (shown above
to be proportional to
ionization by cathode
rays), air = 1.
Ionization by
Rontgen rays.
Ionization of air taken
as unity.
Air
Oxygen
Nitrogen ....
Carbon dioxide . .
Hydrogen ....
Nitrous oxide. . .
100 100
1106 M
•97 89
153 j 14
•069 •as
152 14.7
The numbers, with the exception of those for hydrogen, present an agreement
which is very striking, and show that although the two forms of radiation are so very
different, still the products of their action upon the gases cited are practically the
same.
While the difference in the nimibers for hydrogen is very large, there seems to be
some doubt as to the proper value to be assigned to the conductivity produced by
Rontgen rays in this gas. The conductivities under Rontgen rays in the gases
named have been measured by a number of experimenters, and while their values for
the other gases differ but little, a very wide divergence exists in their numbers for
hydrogen. Rutherford* gives the value '5, while PERRiNf bas obtained the
number '026 by a method entirely different from that of any of the others.
Though we have been thus led to conclude that the density of a gas should
determine its conductivity under cathode rays, strong evidence exists against adopt
ing any such general conclusion regarding the conductivity produced by Rontgen
rays, notwithstanding the general agreement indicated above for the gases cited.
With such gases as HCl, Cl^, SO2, and HgS, J. J. Thomson, Rutherford, and
Perrin have found the conductivities given in Table XI.
From an examination of these values and a comparison with those of Table X., it
is evident that it is quite impossible to deduce any such relation between the densities
of the gases and their conductivities under this radiation.
* * Phil. Mag.,' April, 1897, p. 254
t * Th^se pr^ent^ k la Faculty des Sciences de Paris,* 1897, p. 46.
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TRAVERSED BY CATHODE RAYS. 75
Table XL — Conductivity under Rontgen Rays.
Gas. Deiisitv.
Measurcil by
Pkrrin.
J. J. Thomson. Eithkrford.
I
• HCl 125 .89 11
SO., : 223 6*4 4
i Cl.> 245 174 : 18
, U^S 119 CO 6
8 :
G I
Although the laws of ionization and absorption for cathode rays are clearly defined
by these results, it is difficult to apply them in practice to the direct calculation of
the relative ionizations in any particular experiment.
Take, for example, the case of a pencil of pai*allel rays, 1 sq. centim. in cross
section, traversing air at a pressure p.
Let q = the rate at which ions are produced in 1 cub. centim. of air at unit
pressure by cathode rays of unit intensity
and \) = the coefficient of absorption of air for unit pressure.
Consider then the ionization between two planes distant x and x + c/x, from the
source of the rays.
If I denotes the original intensity of the rays, I . e"**^ will represent their
intensity at a distance x, and p.q .1. c'^^'^'dx wiU then represent the total number of
ions produced between these two planes in one second.
Imagine now a saturating electric field applied at right angles to the rays and
confined between the limits r and r {• d.
The value of the total saturation current obtained with this field would then be
fr+d
p.q.l. e'^^dxy
or i = ^.e^'(l^e^^) (1),
At)
where pX^ is replaced by the quantity X, whose values for different i)ressuro8 are
given in Table IX.
If the air traversed be now subjected to diminishing pressures, the saturation
current will assume different values and will reach a maximum when
i.e., {r+ d)e'"'^r = 0,
. = '• + '' (2).
L 2
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76 MR. J. c. Mclennan on electrical conductivity in gases
An experiment somewhat analogous to this is described, in Section IX. The
apparatus used is shown in figs. 4 and 5. The diameters of the electrodes C and D
were each about 1 centim. and, as already stated, the distance between the window
and the centre of each of the chambers was about 2 centims.
By applying the equation (2) to this experiment, and taking r = 15 centims.* and
d = I centim., it follows that the saturation current would be a maximum when
€^= 166 ...
or X = '5.
From Lenard's values, Table IX., it will be seen that this value corresponds
approximately to a pressure of about 120 millims. of mercury. The observed results,
however, Table II. and fig. 6, indicate a maximum of about 75 centims. Further,
the calculated values of the current from equation (1) exhibit a more rapid rise than
that actually observed.
But the difierence in the results is not surprising. The field within the receiver
was far from uniform, being disturbed by the proximity of the walls of the chamber.
The presence of the narrow tube through which the rays were conducted into the
receiver also produced irregularities. On this account it was impossible to define,
even approximately, the region from which the saturation current was drawn.
Moreover, the actual paths of the rays, as Lenard has pointed out, are largely
influenced by the pressure of the gas traversed. Even at best, then, the calculated
results can scarcely be regarded as more than a rough approximation.
12. Summary of Results,
1. The conductivity impressed upon a gas by cathode rays is similar to that
produced by Rontgen and uranium rays, and can be fully explained on the hypothesis
that positive and negative ions are produced by the radiation throughout the volume
of the gas traversed.
2. When cathode rays are allowed to fall upon insulated metallic . conductors
surrounded by air at atmospheric pressure,
(a.) such conductors if initially uncharged gain a small limiting negative charge,
(6.) positive charges are completely dissipated,
(c.) negative charges drop to a small limiting value,
(o?.) the loss of charge is due to the action of the ionized air surrounding the
conductor, and the value of tlie limiting negative charge is determined by
the extent of the conduction in this air.
3. The ionization produced in a gas by rays coming from the aluminium window
in a Lenard discharge tube is due to cathode rays and not to Rontgen raya
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TRAVERSED BY CATHODE RAYS. 77
4. Lenard's results obtained by fluoroscopic methods on the absorption of cathode
rays are confirmed by a study of the ionization these rays produce in gases.
5. When cathode rays of a given strength are passed through a gas, the number
of ions produced in 1 cub. centim. depends only upon the density of the gas, and is
independent of its chemical composition.
6. With rays of constant intensity the ionization in any particular gas varies
directly with the pressure to which it is subjected.
7. The relative ionizations produced by cathode rays of constant intensity in air,
oxygen, nitrogen, carbon dioxide, hydrogen, and nitrous oxide, at the same pressure,
are expressed by the numbers which represent their densities.
8. With cathode rays, just as with Rontgen rays, the number of ions produced in
a gas bears a definite ratio to the amount of the radiant energy absorbed.
I gladly avail myself of this opportunity to record my grateful sense of the never
failing encouragement and assistance received from Professor J. J. Thomson.
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[ 79 ]
III. Mathematical Contributions to the Theory of Evolution. — VIII. On the Inheri
tance of Characters not capable of Exact Quantitative Measurement. — Part I.
Introductory. Part II. On the Inheritance of Coatcolour in Horses. Part III.
On the Inhentance of Eyecolour in Man.
By Karl Pearson, F.R.S., ivitk the assistance of Alice Lee, D.Sc.y University
College^ London.
Received August 5, — Read November 16, 1899 ; withdrawn, rewritten, and again received
March 5, 1900.
Contents.
Pakt I.— Introductory.
Page
§ 1. General Nature of the Problem and Assumptions upon which it can be solved 80
§ 2. Determination of the Mean Value of the Characters and the Ratio of the Variability of
Correlated Characters 81
§ 3. Determination of the Probable Errors and Error Correlations of all the quantities involved . 82
§ 4. On the Construction of Normal Scales for Characters not capable a prim'i of exact
quantitative Measurement 87
§ 5. On Blended and Exclusive Inheritances 88
Part II. — On Coatcolour Inheritance in tlie Tlwrougldned Horse.
§ 6. On the Extraction and Reduction of the Data 92
§ 7. On the Mean Coatcolour of Thoroughbred Horses 93
§ 8. On the Relative Variability of Sex and Generation 94
§ 9. On the Inheritance of Colour in Thoroughbred Horses 98
(a) Direct Line, First Degree, {b) Direct Line, Second Degree. {c) Collateral
Inheritance. General Conclusions.
Part III. — On Eyecolour Inheritance in Man.
§ 10. On the Extraction and Reduction of the Data 102
§11. On the Mean Colour, having regard to Sex and Generation 104
§ 12. On the Relative Varial)ility of Sex and Generation 109
§13. On the Inheritance of Eyecolour in Man 113
(a) Aflsortative Mating, (b) Collateral Heredity, First Degree, (c) Collateral
Heredity,. Second Degree, (d) Direct Heredity, First Degree, (e) Direct
Heredity, Second Degree. (/) On Exclusive Inheritance.
§ 14. General Conclusions 119
Appendix I. Tables of Coatcolour in Horses 122
II. Tables of Eyecolour in Man 138
Note I. Inheritance of Temper and Artistic Instinct 147
Note 11. On the Correlation of Fertility and Eyecolour 148
VOL. CXCV. — A 264. 29.10.1900
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80 PROFESSOR K. PEARSON AND DR. A. LEE ON
NOTE.
Thifl memoir was originally presented to the Society on August 5, 1899, and read on November 16,
1899. In working out by the same theory the coefficients of inheritance for Basset Hounds, Mr. Leslie
BramleyMoore discovered that the method adopted was not exact enough in its process of propor
tioning. Accordingly, with the assistance of Mr. L. N. G. Filon, we immensely developed the theory, so
that it was necessary to rewrite the theoretical part of the original memoir. This has been carried out in
Part VIL of this series. The present memoir consists substantially of the portions of the original
memoir relating to the inheritance of coatcolour in Horses and eyecolour in Man, with the numerical
details and the resulting conclusions modified, so far as the extended theory rendered this necessary. In
the very laborious work of reconstructing my original tables I have received the greatest possible assistance
from Dr. Alice Lee, and I now wish to associate her name with mine on the memoir.* The memoir
was at my request returned to me for revision after it had been accepted for the * Philosophical
Transactions.*
Part I. — Introductory.
(1.) A CERTAIN number of characters in living forms are capable of easy observation,
and thus are in themselves suitable for observation, but they do not admit of an
exact quantitative measurement, or only admit of this with very great labour. The
object of the present paper is to illustrate a method by which the correlation of such
characters may be effectively dealt with in a considerable number of cases. The con
ditions requisite are the following : —
(i.) The characters should admit of a quantitative order, although it may be
impossible to give a numerical value to the character in any individual.
Thus it is impossible at present to give a quantitative value to a brown, a bay, or a
roan horse, but it is not impossible to put them in order of relative darkness of shade.
Or, again, we see that a blue eye is lighter than a hazel one, although we cannot
d priori determine their relative positions numerically on a quantitative scale.
Even in the markings on the wings of butterflies or moths, where it might be
indefinitely laborious to count the scales, some half dozen or dozen specimens may
be taken to fix a quantitative order, and all other specimens may be grouped by
inspection in the intervals so determined.
We can even go a stage further and group men or beasts into simply two
categories — ^light and dark, tall and short, dolichocephalic and brachycephalic — and
so we might ascertain by the method adopted whether there is, for example, correla
tion between complexion and stature, or stature and cephalic index.
(ii.) We assume that the characters are a function of some variable, which, if we
* I have further to thank Mr. Leslie BramleyMoore, Mr. L. N. G. Filon, M.A., Mr. W. R.
Macdonell, M.A., LL.D. and Miss C. D. Fawcett, B.Sc, for much help in the arithmetic, often for
laborious calculations by processes and on tables, which were none the less of service if they were
afterwards discarded for others. To Mr. BramleyMoore I owe the extraction and part of the
arithmetical reduction of the horsecolour tables.
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 81
could determine a quantitative scale, would give a distribution obeying — at any rate
to a first approximation — the normal law of frequency.
The whole of the theoretical investigations are given in a separate memoir, in
which the method applied is illustrated by numerical examples taken from inheri
tance of eyecolour in man, of coatcolour in horses and dogs, and from other fields.
We shall not therefore in this paper consider the processes involved, but we may
make one or two remarks on the justification for their use. If we take a problem
like that of coatcolour in horses, it is by no means difficult to construct an order of
intensity of shade. The variable on which it depends may be the amount of a
certain pigment in the hair, or the relative amounts of two pigments. Much the
same applies to eyecolour. In both cases we may fail to obtain a true quantitative
scale, but we may reasonably argue that, if we could find the quantity of pigment,
we should be able to form a continuous curve of frequency. We make the assump
tion that this curve — to at any rate a first approximation — is a normal curve. Now
if we take any line parallel to the axis of frequency and dividing the curve, we
divide the total frequency into two classes, which, so long as there is a quantitative
order of tint or colour, will have their relative frequency unchanged, however we, in
our ignorance of the fundamental variable, distort its scale. For example, if we
classify horses into bay and darker, chestnut and lighter, we have a division which is
quite independent of the quantitative range we may give to black, brown, bay,
chestnut, roan, grey, &c.
Precisely the same thing occurs with eyecolour ; we classify into brown and darker,
hazel and lighter, and the numbers in these classes will not change with the
quantitative scale ultimately given to the various eyetints. Our problem thus
reduces to the following one : Given two classes of one variable, and two classes of a
second variable correlated with it, deduce the value of the correlation. Classify sire
and foal into bay and darker, chestnut and lighter ; mother and daughter into brown
and darker, hazel and lighter, and then find the correlation due to inheritance
between the coatcolour or eyecolour of these pairs of relations. The method of
doing this is given in Memoir VII. of this series. Its legitimacy depends on the
assumptions (i.) and (ii.) made above, which may I think be looked upon as
justifiable approximations to the truth.
Of course the probable error of the method is larger than we find it to be when cor
relation is determined from the productmoment. Its value varies with the inequality
of the firequency in the two classes given by the arbitrary division. It will be
least when we make that frequency as nearly equal as possible — a result which can
often be approximately reached by a proper classification. In our present data the
probable errors vary from about "02 to '04, values which by no means hinder us from
drawing general conclusions, and which allow of quite satisfactory general resulta
(2.) So far we have only spoken of the two classes, which are necessary if we
merely want to determine the correlation. But if we wish to deal with relative
VOL, CXCV. — A, M
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82
PEOPESSOR K. PEAESON AND DR. A. LEE ON
variability we must have more than two classes. We have, in fact, in our tables
preserved Mr. Galton's eight eyecolour classes and the seventeen classes under which
the coatcolour of thoroughbred horses is classified in Wethbrby^s studbooks. Such a
classification enables us at any rate approximately to ascertain relative variability,
and, what is more, to reconstruct approximately the quantitative scale according to
which the tints must be distributed in order that the frequency should be normal.
For, in order to attain this result, we have to ascertain from a table of the areas of
the normal curve the ratio of the length of the abscissa to the standard deviation which
corresponds to any given increase of frequency. Let us suppose that three classes
have been made — n^, n^, nj, represented by the areas of the normal curve in the
accompanying diagram so marked. Let pi and p^ be the distances of the mean from
Ox, x^
the two boundaries of n^ Here Pi may be negative, or p^ infinite, &c. Then if
Aj = p^jay A3 = pj(r^ we find at once, if N = total frequency,
"^^ = viiy""^ (>■)•
"■'r'" = a/!D"<^ ()•
Now the integrals on the right are tabulated, and thus, since the lefthand side is
a known numerical quantity, it follows that pja and Ps/o, and accordingly the range
(Ps ~V\)I^ ^^ *^® (AsiSR in terms of the standard deviation, are fully determined.
Thus, if € be the range on the scale of tint or colour of the group of which the
observed frequency is ng, we have € = j^g — p^, and thus c/cr = q say, is known.
For a second series c/cr' = q\ Hence aja = q/q, and accordingly the ratio of the
variabilities of the two series is determined.
Again, the ratio pj{pz — Pi) enables us to find the position of the mean in terms
of the range on the scale occupied by the tint corresponding to the frequency n^.
As a rule we shall take this tint to be that in which the mean actually lies, in which
case we shall have pjipz + P\) as determining the ratio in which the mean divides
the true quantitative range of this particular tint.
(3.) Let 7; = p^{pzPi) = h/{KK) (iii)>
^ = cr/ar' = (VV)/(^3^) (iv.)
It remains to find the probable errors of these quantities.
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 83
Suppose Sj. to be the standard deviation for the errors in a quantity a?, and R,y
the correlation coefficient for errors in two quantities x and y.
Further let
H^;^'" W'
where subscripts and dashes may be attached to H to correspond to like distin
guishing marks attached to h.
Since
2^1^ _ 1 tei^dx hi )
2N v/2^Jo*' "^ ^^^'^
we have at once Snj =: NHjS^i,
and S,. = 2«y(NH0 (vU.).
Similarly, Sn^ = — NH38A3, whence :
24. = W(NH3) (viii.).
Further, we have 2«.2>i,R*.«, =  S,,2..R«.,y(N«HiH3) (ix.) ;
but, as is shewn in Part VII., § 4,
, __ Wi(N«A ^ ss _ n,(Nn,) , .
MA..= ^ • . . (^.).
Thus we find
Probable error of ^1 = *67449Si,
_ 67449 1_ A(Nn,) , •• .
~ v/N Hi V N2~~ ^^"•^•
Probable error of A3 = 'T/W'W A/ "' n~ (xiii.).
S
Correlation in errors in hi and A3, or R^^, is given by
2*.S^RiA=l^ (^^^^
Let u = A3 — Aj, tt' = A3' — A/ be the ratio to the respective standard deviations
of the ranges corresponding to the groups n^ and rij'. Then
~ N»l Hi« "^ H,« HiH,J '
whence, if i/ be a proportional frequency = n/N, we readily find
Prol»bleerrorofu = ^''{^+i(^_ + ^7 . . . (xv,).
Rotable «ror of u'=:^'{l+l(i + ^,)*}'. . . (.vi.).
M 2
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84
PROFESSOR K. PEARSON AND DR. A. LEE ON
I now proceed to determine the correlation in the errors made in determining the
ranges corresponding to any two classes of any two variables which are correlated.
For this purpose let the frequency correlation table be dressed as follows, in the
diagram below.
kAj>i< —
h.k
»
i
Axis of X variable, g ':
'/
*.
TotAL
^
§•
/77„
m^i
"^Sl
n',
\s0
M
^
\
^%y,
>
q''
If
m,g
m^g
fUss
"i
>
1
1
T
s
5&
V^
HsHJ)
fn,s
/»«
Mas
r>^
ToCclLs
",
n.
n^
N
Here m,y denotes the frequency of individuals common to the two classes n, and
riy Let M(/ denote its " conjugate," or all the frequency which appears in neither Ui
nor v!j ; then
N = My + n, + n'j — m^ (xviL).
As before, we have
J ^ n i(N — n,)
N
2_ »t'XN«0)
N
g _ 'MyCN^fy) ^ 2 _ mi,(N  V/iy)
'^ — N ' ^^ — N
(xviii.).
(xix.).
Further, since m^ and M(; are mutually exclusive, we have
Mj/ZWy
From (xvii.) we have for small variations
(XX.).
Hence
2S^X,R.^,= V+VS.* V22.,tH,RM^ . . . (xxi).
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 85
Substituting the values given above we find, after some reductions,
2„.2«..K^,;,^ = —^^ ' (xxu.)
This result, which is extremely simple in form, gives the correlation in errors made
in determining the frequencies in any two classes whatever of any two correlated
variables.
I next proceed to find the correlation between errors in u and u\ the ratio of the
ranges occupied by any two classes to their respective standard deviations.
We have
8n^ = — NH38A3.
Hence S(A,  *,) = ^ + I'd;  i).
Similarly 8(V  A,) = ^. + f (g.  .).
Multiply the first by the second, and summing as usual for all possible errors, we
have, by using (xxii.)
^^p _ 1 JNmaa  71^71^' . Nm^g ' n^n^' / 1 1 \ 'Sm^  n^\ / 1 1\
z«z,.Xi,^  N 1 N^HiH/ " ■•" N»Hi VHi' sj ""^ N^H/ \Ei, ~ hJ
Collecting the like H's we find, after very considerable reductions,
. . . . (xxiiL),
, Z.2,^n,,.  jj I jj^jj^, h jj^jj^, f jj^g^, I jj^jj^, I . . vxxui. ;.
or,
where /ty = wi^/N = proportional frequency.
A glance at our diagram on the previous page of the correlation table divided into
nine classes, shows at once the symmetrical formation of this result. By writing at
the points P, Q, S, and T, the ordinate there of the normal surface, on the supposition
of no correlation and N = 1, the construction of the result is still more clearly
brought out.
We are now in a position to determine the probable errors of 17 and ^ We have
8,,=
__ h^Bh^ — AjS/ij
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86 PROFESSOR K. PEARSON AND DR. A. LEE ON
Hence
 «*N 1 (AiH,)«Na "^ (A8H3)'N2 (A,Hi)(A2H,)N2i '
Or, Probable error of t)
where u is the range A3 — h^, and v^ and 1^3 are the proportional frequencies, as before.
Care must be taken, if the class n^ cover, as it usually will in our present investiga
tions, the mean, to put h^ negative within the radical. In other words, for a class
covering the mean we have
Probable error of rj
 '67449 h,K r_?L . _s_ _ fjH Ys^yV /xxv^
Lastly we have
u* u \u' tt / *
*f ~ »» \ V,'* "^ tt« ««' J •
Thus : Probable error of ^
= 67«.i{'+^^}' (.xvi.),
where we have by (xv.), (xvi.), and (xxiii."*)
2 81 Jiii . jv _/< . ivyi
V < p _ i J /*ii  "i^i' , M »8  ^sV , /*i8  yiV _i_ /^i r "V!!'!
*"^"' """Nl H,Hi' ^"HsH,' "•■ H1H3' +'H3Hi' /'
where, as before, fi'a and v'a represent proportional frequencies.
In the following investigations on coatcolour, and eyecolour inheritance I have
not thought it needful to give in every one of the thirtysix relationships dealt with
the probable errors of the means, ratio of variabilities, and the coefficients of inheri
temce (»;, 4> and r). The arithmetical labour would have been too great, for the
or,
Hence
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MATHEMATICAL CONTEIBUTIONS TO THE THEORY OP EVOLUTION.
87
expressions as given above are somewhat complex. It is, however, necessary to keep
the approximate values of these probable errors in view, and, as our results classify
themselves easily into groups for which our data, as well as the intensity of heredity,
are approximately the same, one series of these errors has been found for each group.
(4.) If we have ground for our assumption that the variable at the basis of our
tint classification can be so selected as to give a normal distribution, we may deter
mine the relative lengths on the scale of that variable occupied by each tint or shade.
Thus if (Ti be the standard deviation of the variable for male eyecolour, og for
female eyecolour, I measured the range on the scale in terms of oj and a^ for
Mr. Galton's eight eyecolour tints for 3000 cases of male and 3000 cases of female
eyecolour. I found the spaces occupied on the unknown scale to be as follows : —
No.
Tint.
Eange in terms of 02.
Range in terms of oi.
1
2
3
4
5
6
7
8
Light blue . *
Blue, dark blue
Grey, bluegreen
Dark grey, hazel
Light brown .
Brown . . .
Dark brown. .
Very dark brown.
black!
00
139276
•73468
•40027
•03893
•43679
•84161
00
00
134918
•77596
•41992
•00856
•35895
•64167
00
These results are not so regular as we might have hoped for, on the assumption
that the ratio of ai/at^ would be the same from whatever part of the scale it be
determined. The general conclusion, however, would be that a^ is slightly larger
than 0*2, which is confirmed by other investigations. Actually a tint may be rather
vaguely described, and where the data were obtained by untrained observers without
the assistance of a plate of eyecolours, a good deal of rather rough classification is
likely to have taken place. I do not think it would be safe to go further than stating
that on the quantitative colour scale the tints as described must occupy spaces in
about the following proportions : —
Light
Blue.
Blue,
Dark Blue.
Grey,
BlueGreen.
Dark Grey,
Hazel.
Light
Brown.
Brown.
Dark
Brown.
Very dark Brown,
Black.
00
137
•75
•41
•02
•40
•74
00
Taking 2000 colts and 2000 fillies, the standard deviations being ai and 03 respec
tively, I have worked out the coatcolour ranges in terms of 03 and cr, for each of
the sixteen colours* occurring in the records. We have the following results ; —
* See p. 92, below.
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88
PROFESSOR K. PEARSON AND DR. A. LEE ON
Tint.
Range in 02.
Range in a^.
Range in oj.
Range in oj.
1
00
ao
9
•00000
•00000
2
•12683
•10768
10
196956
201658
3
•00000
•03313
11
•00000
•00000
4
•91747
11 1055
12
•02490
•00000
5
•00000
•00352
13
•00000
•00000
6
•11059
•10451
14
•00000
•00000
7
1 •34684
1 •27688
15
•00000
•00000
8
•00000
•00000
16
00
00
Here again it seems to me that the most we can safely do is to consider that on a
suitable scale the relative lengths occupied by the classes of coatcolours recognised
by thoroughbred horse breeders would be somewhat as follows : —
bl.
bl./hr.
br./bl.
br.
br./b.
b./br. b.
b./ch.
ch./b.
ch.
ch./ro. ro./ch.
i
ro.
ro/gr.
gr/ro.
g^.
' 00
•12
•02
roi
•00
•11 1^31
•00 ' •oo
199
•00
•01
•00
•00
•00
00
1
The reader must carefully bear in mind that these represent scalelengths occupied
by the coatcolour and not the frequency of horses of these individual coatcolours.
What we are to understand is this : that if eyecolour in man and coatcolour in
horses were measured on such quantitative scales as we have given in skeleton, then
the distribution of the frequency of the several colours would be very approximately
normal. The actual skeleton scales are represented in the accompanying diagram,
which puts them at once before the eye.
br/bC.
T
Normal Scale 0(f Colour Ranges in Thoroughbred Horoes.
tibr. ch/h.
BLsLck
Brown
Bay
Chestnut
smn
tr/b.
^m
chM'Si^.
Normal Scale of Eye Colour Ranges in Man.
am^^^"^'
Brtmr, %S^
Grey
BLue6reen
Odrk blue.
Blue
LighC
BlSe
(5.) It is necessary here to draw attention to a distinction of some importance in
heredity, namely, that between blended and exclusive inheritance. In my treatment
of the law of ancestral heredity,* it is assumed that we have to deal with a quanti
tatively measurable character, and that the ancestry contribute to the quantity of
this character in certain proportions which on the average are fixed and follow certain
definite numerical laws. Such an inheritance is blended inheritance. But another
* *Ro7/Soc. Proc.,' vol. 62, p. 386.
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 89
type of inheritance is possible. We may have one progenitor, prepotent over all others
and absorbing all their shares, who hands down to the of&pring not a proportion of
his character, but the whole of it without blend. If this progenitor is a parent we
have exclusive inheritance, if a higher ancestor a case of reversion. I have dealt at
some length with this type of inheritance imder the title of the Law of Reversion
in another paper.* We must consider in outline the main features of such inheritance,
for the cases of eyecolour in man and coatcolour in the horse approximate more closely
to the numerical values required by it, than to those indicated by the law of ancestral
heredity. The chief feature of exclusive inheritance is the absolute prepotency of
one parent with regard to some organ or character. It need not always be the
parent of the same sex, or the same parent throughout the same family. Some
offspring may take absolutely after one, others after another parent for this or that
organ or character only. I believe Mr. Galton first drew attention, in his * Natural
Inheritance' (p. 139), to this exclusive or, as he terms it, alternative heritage
in eyecolour. In going through his data again I have been extremely impressed by
it ; even those cases in which children might be described as a blend, rare as they are,
are quite possibly the result of reversion rather than blending. If we suppose exclu
sive inheritance to be absolute, and there to be no blending or reversion, it is not hard
to determine the laws of inheritance. Supposing the population stable, onehalf the
oflGspring of parentages with one parent of given eyecolour would be identical with
that parent in eyecolour, the other half would regress to the general population
mean, i.e., the mean eyecolour of all parents. Hence, taken as a whole, the regression
of children on the parent would be '5. In the case of the grandparent the regression
would be '25 ; of a great grandparent '125, and so on. With an imcle a quarter of
the oi&pring of his brother will be identical in eyecolour with him, the other three
quarters will regress to the population mean, thus the regression will be '25. If we
have n brethren in a family, and take all possible pairs of fraternal relations out of it,
there will be \n{n — 1) such pairs ; ^ brothers will have the same eyecolour that of
one parent, the other \n brother that of the other parent. Hence selecting any one
brother, \n — 1 would have his eyecolour, and on the average \n would have
regressed to the mean of the general population. In other words, the coefficient of
regression would be {\n — l)/(Jn ^ 1 + ^) = (^n — l)/(w — 1).
Accordingly
«= 3
Begressiou
= 25
n = 4
= 3333
«= 47
= 3649
» = 5
= 375
n = 53
= 3833
n = 6
= '4
n = 00
= '5
♦ *Koy. JSoc. Proc./ vol. 66, pp. 140 d setj.
VOL. CXCV. — A. N
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90 PROFESSOR K. PEARSON AND DR. A. LEE ON
It will at first appear, therefore, that the fraternal regression with the size of
families actually occurring will vary from '35 to '4.
To some extent these values would be modified by assortative mating, which
actually exists in the case of eyecolour. The correlations between parent and
offspring and between brothers would both be slightly increased. Thus if p be the
coefficient of assortative mating, Ry the fraternal correlation with and Vf without
assortative mating, and r the coefficient for parent and offspring,*
If we put Vf = '36, ?' = '5, /o = '1, we find
R/ = 39.
Thus we see that the regression or correlation for fraternal inheritance in the case
of exclusive inheritance could not, with the average size of families, be very far from
'4 of blended inheritance.
A further source which can modify immensely, however, the fraternal correlation is
the prepotency of one or other parent, not universally, but within the individual family.
In the extreme case all the offspring might be alike in each individual family. Thus
fraternal correlation might be perfect although parental correlation were no greater
than 5. Hence, where for small families we get a fraternal correlation greater than
•4 to '5, it is highly probable that there exists either a sex prepotency (in this case,
one of the parental correlations will be considerably greater than the other) or an
individual prepotency (in which case the parental correlations based on the average
may be equal). We shall see that fraternal correlations occur greater than '5 in our
present investigations. I have dealt with these points in my Memoir on the * Law of
Reversion,'! and also in the second edition of the ' Grammar of Science. 'J
Another point also deserves notice, namely, that with the series '5, '25, '125, &c.,
for the ancestral coefficients in the direct line, the theorems proved in my Memoir on
Regression, Heredity, and Panmixia§ for the series of coefficients ?% r^, r^ . . . exactly
hold. That is to say, if we have absolutely exclusive inheritance, the partial regres
sion coefficients for direct ancestry are all zero except in the case of the parents.
This it will be observed is not in agreement with Mr. Galton's views as expressed in
Chapter VIII. of the * Natural Inheritance.' But I do not see how it is possible to
treat exclusive inheritance on the hypothesis that the parental regression is about '3.11
Actual investigation shows that for this very character, i.e., eyecolour, it is nearer '5.
If we take Table XIX. of Mr. Galton's appendix, and make the following groups, both
* This is shown in a paper not yet published on the inj9uence of selection on correlation.
+ *Roy. Soc. Proc./ vol. 66, pp. UO et seq„
t " On Prepotency," p. 459 ; " On Exclusive Inheritance," p. 486.
§ ^Phil. Trans.,' A, vol. 161, p. 302, etc.
II Mr. Galton takes I throughout his arithmetic.
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MATHEMATICAL C50NTRIBUTI0NS TO THE THEORY OF EVOLUTION.
91
parents light, one parent light and one medium, one light and one dark, we reach the
following results : —
Parents* eyecolour.
Children, actual.
Lighteyed children, calculated.
Total.
l'oHy^i.: iJSJl
Ancestral law with
knowledge of parents
and grandparents.
Both light. . . .
Light and medium .
Light and dark . .
355
215
211
334 355
170 161
107 105
1
321
160
117
1
Here the exclusive inheritance leads us to misplace thirtytwo and the ancestral
law thirtythree children. The evidence, therefore, of the correctness of the latter is
hardly greater than that of the former. Indeed, if the former were modified for
reversion, it would very possibly give better results than the latter.
I am inclined accordingly to look upon eyecolour inheritance as an exclusive
inheritance modified by reversion, and, to some extent, by assortative mating, rather
than a mixture of exclusive inheritance with a slight amount of blending. In either
case exclusive inheritance gives results like the above so closely in accord with the
ancestral law that the latter might be supposed to hold. But, theoretically, I do not
understand how the ancestral law is compatible with exclusive inheritance.
Theoretically, we have values of parental, avuncular, and grandparental correlation
greater than the ancestral law would permit of, and these theoretical values are, on
the whole, closer to observation, as we shall see in the sequel, than those given by the
law of ancestral heredity. The following table gives the two systems : —
Table I. — Theoretical Values of the Regression Coefiicients.
Belationship.
Blended inheritance,
ancestral law.
Exclusive inheritance,
absolute, no reversion.
Parent and offspring
Qrandparent and offspring . . .
Greatgrandparent and offspring .
Brethren
Uncle and nephew
•3
•15
•075
•4
•15
•5
•25
•125
•36 to 5*
•25
Now, if exclusive inheritance be modified by reversion or assortative mating, or if
blended inheritance be modified by " taxation," t then we shall get values. different
* This varies with the size of the family,
t * Roy. Soc. Proc.,' vol. 62, p. 402.
N 2
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92 PROFESSOR K. PEARSON AND DR. A. LEE OK
from the above, and possibly filling up the numerical gap between them. To this
point I shall return when dealing with the observed values for eyecolour in man.
Part II. — On ColourInheritance in Thoroughbred Racehorses.
(6.) All the data were extracted from Weatherby's studbooks, the colours being
those of the horses as yearlings. My first step was to form an order, not a quantita
tive scale, of horsecolours. With this end in view, the recorded colours were
examined, and, including the arabs, the following seventeen colours were at first
found : —
1. Black (bl.).
2. Black or brown (bl./br.).
3. Brown or black (br./bl.).
4. Brown (br.).
5. Brown or bay (br./b.).
6. Bay or brown (b./br.).
7. Bay (b.).
8. Bay or chestnut (b./ch.).
9. Chestnut or bay (ch./b.).
10. Chestnut (ch.).
11. Chestnut or roan (ch./ro.).
12. Roan or chestnut (ro./ch.).
13. Boan (ro.).
14. Roan or grey (ro./g.).
15. Grey or roan (g./ro.).
16. Grey(g.).
17. White (w.).
Now, if we take the alternative colours to mean that the first alternative is the
prominent element, we see that these colours in use among breeders admit of only one
arrangement from black to white. That is to say, that a continuous shadechange is
actually in use for the colournomenclature of thoroughbred horses.* Thus without
any hypothesis as to the quantitative relative values of bay or roan, we have an order
which serves for all our present purposes. Following this order. Appendix I., Tables
I. — XII., for the colour correlation of related pairs of horses was compiled by
Mr. Leslie BramleyMoore from the studbooks. When dealing with relationship
m the ? line only, no weight has been given to fertility, as each mare has had only
one foal attributed to it, or two in the case of fraternal correlation. In the case
of the c? line, the colours of the older sires were harder to ascertain, and we did not
obtain altogether more than 600 sirecolours. Thus one, two, or, in a few cases, three
or four colts or fillies were taken from each sire.
I shall now discuss the results which may be drawn from these tables for the theory
of heredity, first placing in a single table all the numerical constants calculated from
the data in Tables I. to XII. of Appendix I.
^ Among the 60008000 horses dealt with only four were found with colours not entered in this scale,
but these entries of bl./ch., br./ch., b./ro., in no way contradict the order of the scale, but merely show a
rougher appreciation on the part of the nomendator, or possibly printers' or editor's errors.
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MATHEMATICAL CONTBIBUTIONS TO THE THEORY OF EVOLUTION.
93
Table II. — Coatcolour Inheritance in Thoroughbred Horses.
Pair of relatives.
Division of
bay range by
the mean
Ratio of
variabilities.
Coeffi
cients of
correlar
tion.
Coefficients of
regression.
Num
ber of
cases.
X.
y
Vx
Vv
C =
<ryl<rg.
rxg.
Rw
Rv«.
N.
Sire
Colt. .
FiUy .
Colt. .
Filly .
•6111
•6061
•5359
•5565
•5713
•5719
•6027
•6051
•8712
•8298
•9500
•9036
11478
12051
1^0526
11067
•4913
•5422
•4862
•5668
•4280
•4499
•4619
•5122
•5639
•6534
•5118
•6273
1300
1050 I
1000 !
1000
Sire
Dam
Dam
Maternal
Maternal
grandsire
grandsire
Colt. .
Filly .
•6583
•6359
•5867
•6042
•7030
•7678
r4225
13024
•3590
•3116
•2524
•2392
■5107
•4058
1000
1000
Colt .
Colt .
Filly .
Filly .
Filly .
FiUy .
'(Half
(Whole
(Half
(Whole
(Half
(Whole
Colt. .
siblings)
Colt, .
siblings)
Filly .
siblings)
Filly .
siblings)
Colt. .
siblings)
Colt. .
siblings)
•5908
•5620
•5665
•5684
•5633
•5410
•5908
•5620
•6665
•5684
•5865
•5711
1
1
1
1
•9607
•9555
1
1
1
1
1^0409
1^0466
•3551
•6232
•4265
•6928
•2834
•6827
•3551
•6232
•4265
•6928
•2723
•5568
•3551
•6232
•4265
•6928
•2960
10466
2000
2000
2000
2000
1000
1000
In this table R,y = r^^^cxlcy, 'Ryx = ^jryOy/cr^r. Halfsiblings* are those having the
same dam, but different sires. Further, 17 is measured from the brown end of the bay
range up to the mean.
(7.) On the Mean CoatColour of Horses. — If our theory be correct, this colour
will not differ much from the median colour, and we notice at once a secular change
going on. We have the following order : —
Maternal grandsire of colt t; = '6583
Maternal grandsire of filly = '6359
Sire of colt = 6111
Sire of filly = 6061
Colt (mean value of seven series) . . . = '5816
Dam of colts t; = '5359
Dam of fillies = 5565
Fillies (mean value of seven series) . . = *575S
* I have introduced this expression in my paper on "The Law of Reversion," *Roy. Soc. Proc./ vol. 66,
as a convenient expression for a pair of offspring from same parents whatever be their sex.
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94 PROFESSOR K. PEARSON AND DR. A. LEE ON
Now the colours of all the horses are returned when they are foals, so that there is
no question of any variation of colour with age, yet we notice that —
(i.) The horse is lighter in colour than the mare.
(ii.) If we go back two generations (grandsire) the horse is lighter than if we only
go back one generation (sire), and the sires are again lighter than their colts.
In other words, there seems a progressive change towards a darker coat.
(iii.) On the other hand, the mares one generation back appear to be darker than
their daughters.
(iv.) The average sire of colts is lighter than the average sire of fiUies ; the average
dam of colts is darker than the average dam of fillies.
Now these conclusions seem to indicate that the older horse was lighter in coat, and
the older mare darker in coat than either the colt or filly of today, and that there is a
tendency in the thoroughbred racehorse of today to approach to an equality of colour
in the two sexes, an equality which is a blend of the sensibly divergent sexcolour of
the older generation.
Whether this secular change is due to the " breeding out " of the influence of light
Arabian sires, or to a tendency in the past to select lightcoloured sires and dark
coloured mares for breeding, or to the fact that such coloured sires and mares are the
most fertile, i.e., to an indirect effect of reproductive selection, is not so easy to
determine. But what does appear certain is that the average thoroughbred is
approaching to a blend between its male and female ancestry, which were sensibly
divergent.*
(8.) On the Relative Variability of Sex and Class in Horses. — The following table
gives the length of the bay range in terms of the standard deviation for each class.
If c represent this range, then in terms of the previous notation c = w x <r = ?/ X o^,
and from these values of it and it' the ratio, C = <^/<^' of Table II. was calculated.
* Mean of dams and sires of colts = 5735, «.€., i(*6111 + 5359).
Mean of dams and sires of fillies = *5813, i.e,, ^(*6061 + 5565).
These are curiously enough almost exactly equal to the mean values '5753 and '5816 obtained for fillies
and colts. This inverse relationship is too close to the probable errors of the quantities under investiga
tion for real stress to Ijc laid on it, but it may still turn out to be suggestive.
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION.
95
III. — Table of Bay Eanges.
Relative Pair.
Bay Ranga
Probable Error of Median.
X.
y
U X a,.
u' X ay.
X.
y
Sire
Sire
Dam
Dam
Colt ....
Filly ....
Colt ....
Filly ....
l46943<r«
l64075<ra,
l36645<r,
l38165<ra
l28019(ry
l36I49<ry
l298I9<ry
l24845(ry
± 0160
±•0159
±•0196
±0193
± 0183
± ^0192
±•0206
±•0214
Maternal grandsire .
Maternal grandsire .
Colt ....
Filly ....
l69694<r,
1 650210^,
M9293<r„
l26702<r„
±•0158
±0162
± ^0224
±0211
Colt
(Half
Colt
(Whole
Filly
(Half
Filly
(Whole
Filly
(Half
FUly
(Whole
Colt ....
siblinin)
Colt . . . .
siblings)
Filly . .
siblings)
Filly ....
siblings)
Colt ....
siblings)
Colt ....
siblings)
l23953<ra,
l27688(r«
l39619(ra,
l34684<r,
l33479<r,
1 415010^,
l23953<ry
l27688<ry
l39619<ry
l34684<ry
l28229<ry
l35207(r„
±0153
±•0148
±0135
±•0140
±0202
±•0189
±•0153
±0148
±•0135
±•0140
±0208
±0198
To explain the last double column I note that Mr. Shbppabd has shown (* Phil.
Trans./ A, voL 192, p. 134) that the probable error of the median equals
•84535 o/\/N.
Hence in terms of the bay range we have
probable error of median .q^kqk// /m\
length of bay ranee /\ v /•
length of bay range
I have found that this simple result gives a value close to but slightly larger than
the probable error of the quantity tj (p. 82), from which I have determined the
position of the mean in the bay range. It is much easier to calculate, but of course
not so exact, as we take no accoimt of possible errors in the bay range itself or
their correlation with errors in the median. I have accordingly tabulated its values
in the last double colunm as a rough guide to the errors made in the numbers
upon which the statements in the previous section depend. I shall return to the
consideration of the probable errors below. Turning to columns 3 and 4 of our
Table 11. , we can draw the following conclusions as to the variability of sex and
class: —
(a.) The Younger Generation is more Variable than the Old. — Thus, foals are more
variable than their sires, and, looking at the expressions in Table III. for the bay range,
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96 PROFESSOR K. PEARSON AND DR. A. LEE ON
sii'es than grandsires. This is a rule I have now found true in a very great number
of cases of inheritance. Parents are a fairly closely selected body of the general
population, and so less variable than that population at large. This might appear
pretty obvious in the case of thoroughbred horses when we are dealing with sires and
grandsires. I have already pointed out that it was impossible to take 1000 to 1300
colts or fillies with as many independent sires, the fashion in sires is too marked ; and
of course the number of independent grandsires was still fewer.* But even in the
case of dams, where we have taken as many independent dams as fillies, we see this
reduction in variability in the older generation. As it also occurs with stature, &c.,
in man as well as with coatcolour in horses — in which latter case we expect artificial
selection — it deserves special consideration. Without weighting with fertility, there
exists a selection of the individuals destined to be parents in each generation. We
have to ask whether the change in mean and variability from parent to oflGspring in
each generation is secular or periodic, or to what extent it is partly one and partly
the other. The importance of settling this point is very great ; it concerns the
stability of races. So far as my fairly numerous series of measurements yet go,
I cannot say that a " stable population " has definitely shown itself; the characters of
each race when measured for two generations seem to vary both in mean and
standard deviation. Palseontologists tell us of species that have remained stable for
thousands of years, but this is a judgment hitherto based on a qualitative apprecia
tion. A quantitative comparison of the means, variabilities, and correlations of some
living species in its present and its fossil representatives would be of the greatest
interest and value. For myself, I must confess that my nmnerical investigations so
far tend to impress me with the unstable character of most populations.
(6.) There is fairly good evidence that the horse is more variable than the 7na/re in
coatcolour. It would be idle to argue fi:om the first four results of Table III. that
the mare is more variable than the horse, in that these results show the dam to be
more variable than the sire. For, as we have shown, the process of breeding and our
method of extracting the data tend to produce a much more intense selection of sires
than of dams. But if we compare the mean bay range in terms of the standard
deviation of colts for our seven series of colts with that for the seven series of fillies
in Table III., we find for the first 1*27458 <r<. and for the second 1 '33854 c/. Hence
we are justified in concluding that ac is greater than oy; In fact in only one case out
of the seven does the series of fillies give a less variability than the corresponding
series of colts, i.e., colts corresponding to dams are somewhat less variable than fillies
corresponding to dams. It must, however, be remembered that this conclusion is
based upon the coatcolour of the animals recorded as yearling foals, t Thus, the coat
* For some account of the extent of in and in breeding in the thoroughbred horse, see my memoir on
" Reproductive Selection," • Phil. Trans.,' A, vol. 192, p. 267 et seq,
t The reader must always bear in mind that when we speak of the variability of colour in sire or
dam, <^c., it means the varialiility of this class when they were yearlings.
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION.
97
colour may change both in intensity and variability with age, much as variability in
stature changes with children from birth to adult life.
(c.) As a more or less natural result of (6) it follows that any • group, male or
female, having nutle relatives is more variable than the same group with female
relatives. Thus sires of colts are more variable than sires of fillies ; fillies halfsisters
to colts are more variable than fillies halfsisters to fillies, &c. But out of the nine
cases provided by our data there are three exceptions to the rule, and perhaps not
much stress can be laid on it, at iany rate in the above form. It would appear that
males, relatives of males, are sensibly more variable than males relatives of females.
The bay ranges are 1*3926 </ and 1*4447 a respectively, which indicates that the
average cr' is larger than a. But if we treat the groups of females alone, we find for
females with male relatives the bay range = 1*3694 <r, and for females with female
relatives 1*3433 <t\ which indicates that the latter are more variable. The difference
is, however, not very sensible. Possibly the rule is simply that extremes tend to
produce their own sex, but our data are not sufficient for reaUy definite conclusions on
the point.
In order that we may have a fair appreciation of the probable errors of the
quantities involved and the weight that is to be laid upon their differences, I place
here a table* of the probable errors of ij, of £ = Cx/ay and of r,^ for t3rpical cases.
IV.— Table of Probable Errors.
Belations.
V'
Vy
C
u
u'
ray
Sire and Filly . .
Grandsire and Colt
Colt and Colt . .
(Whole siblings)
Filly and Colt . .
(Half siblings)
•0U3
•0143
•0186
•0179
•0170
•0199
•0186
•0185
•0243
•0237
•0315
•0363
•0385
•0328
•0335
•0330
•0319
•0328
•0328
•0288
•0333
•0259
■0363
It will be seen from this table that the probable error in 17 is about 3 per cent., in
C about 2 to 4 per cent., in u about 2 to 2*5 per cent., and the values of r about *03,
growing somewhat larger as r grows smaller. The probable errors are thus some
what larger than those which we obtain by the old processes when the characters are
capable of quantitative measurement, but they are not so large as to seriously affect
the use of the new processes in biological investigations. As we have already
indicated, the probable errors of the tj'b may be roughly judged by Mr. Sheppard's
formula for the median (p. 95).
It will be seen that the differences in the rfa and fs of Table II., or the u'q of
* I have to thank Mr. W. R. Macponell for friendly aid in the somewhat laborious arithmetic
involved in calculating these probable errors.
VOL. CXCV. — A. O
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98 PROFESSOR K. PEARSON AND DR. A. LEE ON
Table III., are as a rule larger than the probable errors of the diflferences, sometimes
several times larger. Yet in some cases they are not such large multiples of the
probable errors of the diflferences that we can aiSbrd to lay great stress on the
divergence of ly or £ or tt in a pair of special cases. We must lay weight rather on
the general tendency of the tables when all the series are taken together. Thus,
while we may have small doubt about the correctness of (i.) of § 7 or (6) of § 8, we
should look upon (iv.) of § 7 as an important suggestion which deserves serious
consideration rather than a demonstrated law. The same again holds good for (c) of
§ 8. It is because of their suggestiveness that they are here included. That a
differential fertility or an individualisation in the sex of offspring should be corre
lated with colour, would, if proved, be a fact of very considerable interest. It would
again emphasise the important part which genetic selection plays in the modification of
characters.* A priori it is not more unreasonable to expect coatcolour in horses than
to suppose haircolour in men to be correlated with fertility. But the fertility of man
does seem to vary from the light to the dark races. The special feature of interest
here, however, is that a different colour in the two sexes appears to influence the
preponderance of one or other sex in the offspring. It would be an interesting
inquiry to determine whether the sexratio in the oflfepring of " mixed marriages "
varies when the races of the two parents are interchanged.
(9.) On the Inheritance of Coatcolour in Thoroughbred Horses. — (a.) Direct Line.
First Degree, — Having regard to the probable errors — ^about '03 — in the values of
the correlation coefficient r^yy it seems quite reasonable to suppose that the mean
parental correlation, actually '5216, is practically '5. It is quite impossible to
imagine it the '3 of Mr. Galton's view of the Law of Ancestral Heredity. If we
adopt the view of that law given in my paper on the Law of Ancestral Heredity, t
and take the coefficient y to be different from unity, then it is shown in my paper on
the Law of Reversion^ that we cannot on the theory of blended inheritance get
parental correlation as high as '5 without values of the fraternal correlation which
are much higher than those hitherto observed, certainly much higher than, as we
shall see later, we find in the case of coatcolour in horses. Coatcolour in horses does
not thus appear to be at all in accord with Mr. Galton's view of ancestral inheritance,
or even with my generalised form of his theory. It does accord very well with what
we might expect on the theory of exclusive inheritance as developed above, p. 91,
on the assumption that there is no reversion.
Looking at the matter from the relative standpoint, we see that not much stress can
be laid on the respective influences of the sire and dam on the colt, or of the sire and
dam on the filly ; but, on the other hand, the filly does appear to inherit more from
* See the concluding remarks in the memoir on " Genetic (Reproductive) Selection," * Phil. Trans.,'
A, vol. 192, pp. 257—330.
t • Roy. Soc. Proc.,' vol. 62, p. 386 et seq.
J 'Roy. Soc. Proc.,' vol. 66, p. 140 et seq.
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 99
both parents than the colt does. There is certainly (judged from coatcolour) no
preponderance of the sire's influence over the dam's such as breeders appear occasion
ally to imagine. The average influence of the dam on the ofispring indeed appears
to be slightly greater than that of the sire, but the difference is of the order of the
probable error, and not of the overwhelming character exhibited in the case of Basset
Hounds. There is indeed in the case of thoroughbred horses not the same chance of
carelessness produced by a misalliance afterwards screened by the defaulter. There
exists, however, a far greater premium — considering the great value of yearlings from
fashionable sires — set upon dishonesty. Again it is possible that when stallions
receive too many public or private mares, or are still used in their old age, that they
may, without losing the power of fertilising, lose some of the power of transmitting
their character. The divergences, so far as the probable errors are concerned, are
not such that we are forced out of our way to explain them. With the single
exception of sire and colt we see that our table shows the universal prevalence of the
rule that :
Relatives of the same sex are more closely correlated than relatives of the same
grades of the opposite sex. Thus : —
A colt is more like his sire than his dam.
A filly is more like her dam than her sire,
A dam is more like her filly than her colt.
A grandsire is more like his grandcolt than his grandfilly.
A colt is more like his brother colt than his sister filly.
A filly is more like her sister filly than her brother colt.
the latter two cases being true for both whole and half siblings.
The solitary exception is that a sire is more like his filly than his colt.
If we were to assume it a rule that a filly in the matter of coatcolour has stronger
inheritance all round than a colt, we should find it agree with our results for parental
inheritance, and receive considerable support for the much stronger correlation of
fillies than of colts, when either whole or half siblings. But it would not be in
accordance with our results for grandparents, for which, however, we have only
details for two out of the eight possible cases. On the whole, I think we must
content ourselves with the statements that parental correlation is certainly about '5,
and that with high probability each sex is more closely correlated with its own sex of
the same grade of relationship.
(6.) Direct Line, Second Degree. — My data here are unfortunately only for two
cases out of. the possible eight. I hope some day to finish the series, but the labour
of ascertaining from the studbooks the coatcolour of 700 or 800 separate sires is
\rery great. Indeed it is not easy to foUow up the pedigree through the male line
when the sire is not one of the famous few. On the other hand, it is much easier
through the female line. For this reason the maternal grandsire was taken. Even
O 2
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100 PROFESSOR K. PEARSON AND DR, A. LEE ON
in this case we had to seek back for each sire — the year of whose birth was unknown
— ^until we found the record of his coatcolour given under the heading of his dam in
the year of his birth.
The average of our two cases gives a coefficient of correlation == '3353, the colt
having a greater degree of resemblance to the grandsire than the filly. This value is
substantially greater than the '25 we might expect for exclusive inheritance, and
more than double the value '15, to be expected for the grandparental correlation
with Mr. G Alton's unmodified law for blended inheritance. Of course the '25 is to be
expected as the mean of the eight grandparental series, and, as we shall see for eye
colour in man, these may vary very much in magnitude. But allowing for this, it
seems quite impossible that the average value could be reduced to '15. I take it
therefore that the grandparental, like the parental, data point to a law of inheritance
other than that which has been described in my paper on the Law of Ancestral
Heredity. This peculiar strengthening of the grandparental heritage has already
been noted by me in my paper on the Law of Reversion,* and the difficulties of
dealing with it on the principle of reversion therein discussed. There may be some
opinion among breeders as to the desirability of emphasising the dam's strain in the
choice of a sire which leads to this result, but if so it is unknown to me, nor do I see
how it would work without also emphasising the correlation of the dam and foaL
The mean value of the correlation for the maternal grandfather and grandchildren for
eyecolour in man is '3343 — a. result in capital agreement with that for coatcolour in
horses. In that case the average of the eight series, as we shall see later, is con
siderably above '25, and we must, I think, suspend our judgment as to whether this
could possibly in the case of horses be the true mean value. As to the value '15
it seems quite out of the question.
As already remarked, the influence of the maternal grandsire (unlike that of the
sire) is substantially greater on the colt than on the filly.
(c.) Collateral Heredity, Fi7'st Degree. — Here we have more ample data to go
upon, namely, a complete set of six tables of both whole and half siblings of both
sexes.
We notice one or two remarkable features straight off. In the first place, in the
case of both fillies and colts, the whole siblings of the same sex have not a correlation
the double of that of the half siblings, but have a correlation very considerably less
than this. A priori we might very reasonably expect the one to be the double of
the other. This is what would happen in the case of blended inheritance on the
hypothesis of equipotency of the parents. As the half siblings are on the dam's side,
we might assert a considerable prepotency of the dam over the sire. This cannot
indeed be the explanation of the divergence in the case of Basset Hounds, where the
half siblings have a correlation considerably less than half that of whole siblings, t
* *Roy. Soc. Proc.,' vol. 66, p. 140 et seq.
t * Roy. Soc. Proc.,' vol. 66, p. UO dseq.
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 101
and yet the prepotency of the dam in coatcolour is very marked. But in the
present case there is on the average only a slight, if indeed it be a real, prepotency of
the dam. Further, if we turn to the correlation, no longer of siblings of the same
sex, but of opposite sexes, we find the correlation of the whole siblings is approxi
mately double that of the half siblings, as we should d priori have expected.
Taking averages on the assumption that the correlation for whole siblings should
be double that for half siblings, we have the following results : —
Correlation between colts based on results for whole and half siblings . '6667
Correlation between fillies based on results for whole and half siblings . '7729
Correlation between filly and colt based on results for whole and half
siblings '5747
Mean correlation of siblings based upon all results for whole siblings . '6329
Mean correlation of siblings based upon all results for half siblings . . '7100
Mean correlation of siblings based upon results for both whole and
half siblings '6714
We can draw the following conclusions : —
(i.) In whatever manner we deduce the fraternal correlation it is very much larger
than the '4 for whole brethren, or the '2 for half brethren, required by the unmodified
Galtonian law. Such values, as we see above, could be deduced fiom the modified
Galtonian law by taking y greater than unity,* but this would involve values for
the parental correlation sensibly less than those given by theory. We are again
compelled to assert that the modified as well as the unmodified theory of blended
inheritance, based on the Galtonian law, does not fit the facts. The above values,
however, are quite compatible with the theory of exclusive inheritance on the
assumption that there is an individual (not a sexual) prepotency from one pairing to
another.
(ii.) In whatever way we consider it, it would appear that the average value of
the fraternal correlation, as deduced fi:om siblings with the same dam only, is
greater than that deduced from siblings with both the same dam and the same sire.
I am not able to explain this in any way, for I cannot assert a very substantial
prepotency of the dam. All I can say from the data at present available is that for
horses and dogs there appears to be no simple numerical relation between the correla
tion of whole and half brethren.
(iii.) Offspring of the same sex are more alike than offspring of opposite sexes.
This appears to be generally true, so far as our data at present extend, and will be
fairly manifest fi^om the table below.
* 'Boy. Soc. Proc.,' vol. 66, p. 140 e^ seq.
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102
PROFESSOR K. PEARSON AND DR. A. LEE ON
Table V. — Collateral Heredity.
1
i
Pair.
!
Man.
Dog.
Horse.
Stature.*
Cephalic
Index.!
Eyecolour. J
i
Coatcolour.§
1
Coatcolour.
Whole
Siblings.
HaH
Siblings.
BrotherBrother . i
SisterSister . .
BrotherSister . :
•3913
•4436
•3754
•3790
•4890
•3400
•5169
•4463
•4615
i} 5257 1
•6232
•6928
•6827
•3561
•4265
•2834
It will be noted that, with the single exception of eyecolour in man, sister and
sister are more alike than brother and brother.
The mean value of the fraternal correlation for stature is '4034, and for cephalic
index '4027. These results are in excellent accordance with the '4 required by the
Galtonian theory of blended inheritance. The mean values for eyecolour in man,
coatcolour in dogs, and coatcolour in horses are : '4749, '5257, and '6329. These
are quite incompatible with that theory. I venture accordingly to suggest that we
have here cases of an inheritance which does not blend, and that it is an inheritance
which is far more closely described by the numbers we have obtained on the theory
before developed of exclusive inheritance than by the law of ancestral heredity.
Taking in conjunction with these results for collateral heredity, those for parental
and grandparental inheritance, we see that coatcolour in horses diverges widely from
the laws which have been found sufficient in the cases of stature and cephalic index
in man. The latter characters are really based on a complex system of parts, while
the determination of coatcolour may depend on a simple or even single factor in the
plasmic mechanism. Here Mr. Galton's suggestion of an exclusive inheritance of
separate parts ('Natural Inheritance,' p. 139) may enable us to understand why
stature and cephalic index differ so widely in their laws of inheritance from coat and
eyecolours.
Part HI. — On the Inheritance of EyeColour in Man.
(10.) 071 the Extraction and Reduction of the Data. — The eyecolour data used in
this memoir were most generously placed at my disposal by Mr. Francis Galton.
They are contained in a manuscript wherein, by a simple notation, we can see at a
* Pearson, * Phil. Trans.,' A, vol. 187, p. 253 et seq. See Note I. at the end of this paper.
t Pawckti and Pearson, * Roy. Soc. Proc.,' vol. 62, p. 413 et seq,
X Present memoir, p. IIS et seq,
§ Pearson, *Roy. Soc. Proc.,' vol 66, p. 140 et seq.
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 103
glance the distribution in eyecolour of a whole family in its numerous male and
female lines. Such complete details of the various direct and collateral relationships
I have not hitherto come across, and from them I was able to form, in the course of
some months of work, the twentyfour tables of correlation which are given in
Appendix II. These tables, for the first time, give the whole eight series of grand 
parental and the whole eight series of avuncular relationships, besides such as we have
deduced for other characters previously, i.e., the four parental, the three fraternal rela
tionships, and the table for assortative mating. The very great importance of this
material will at once be obvious, and I cannot sufficiently express my gratitude to
Mr. Galton for allowing me to make free use of his valuable data.
At the same time we must pay due regard to the limitations of this material, which
it is needful to enumerate, so that too great stress may not be laid on the irregularities
and divergences which arise when we attempt to reduce the results to laws. These
limitations are as follows : —
(a.) While the data of about 780 marriages are given in the record, they belong
to less than 150 separate families. All our relationships, therefore, contain pairs
weighted with the fertility of the individual families. Thus it was necessary to enter
every child of a mother, every nephew of an uncle, and so forth. In the horse data
we could take 1000 distinct mares and give to each one foal only. That is not possible
in the present case.
(6.) The colour of eyes alters considerably with age. It is not clear that some of
the eyecolours are not given for infants under twelve months, and certainly the eye
colours in the case of grandparents and others must have been taken in old, perhaps
extreme old, age. In a large number of other cases of great grandfather, great great
grandfather, &c., great uncles, and so forth, the eyecolours must have been given
from memory or taken from portraits — in neither alternative very trustworthy sources.
Thus while the horse colour is always given for the yearling foal by the breeder, the
eyecolour is given at very different ages, and comes through a variety of channels.
(c.) The pqysonal equation in the statement of eyecolour, when the scale contains
only a list of tintnames is, I think, very considerable. The issue for the collection of
data of a plate of eyecolours like that of Bertrand would be helpful, but we can
hardly hope for a collection of family eyecolours so comprehensive as Mr. Galton's
to be again made for a long time to come.
These causes seem to me to account for a good deal of the irregularity which appears
in the numerical reduction of the results, but they are not, I hold, sufficient to largely
impair the very great value of Mr. Galton's material.
In tabulating the data, I have followed the scale of tints adopted by Mr. Galton,
and I have used the entire material available in the cases of the grandparental,
avuncular, and marital relations. I nearly exhausted the data for the parental
relationships, but in these tables, which were first prepared, I stopped short at 1000
for the sake of whole numbers. I found, however, that it did not make the arithmetic
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104 PROFESSOR K. PEARSON AND DR. A. LEE ON
sensibly shorter, and I afterwards dropped this limitation. In the case of brethren
I took 1500 of each case — I daresay I could have got 2000 out of the records. As
the lighteyed brethren are entered ^r^^ in Mr. Galton's MS., the First Brother
in my unsymraetrical tables is always lightereyed than the Second Brother, hence
the tables had to be rendered symmetrical by interchanging and adding rows and
columns before we could reduce them. Thus the symmetrical tables have an apparent
entry of 3000 pairs. Of course 1500 is the number used in finding the probable
error of the correlation coeflficient. The like difficulty does not occur in the brother
sister table, where indeed the difference of mean eye colour for the two sexes would
not allow of our making the table symmetrical. A comparison of the symmetrical
with unsymmetrical tables for coltscolts and filliesfillies, will show how little need
there is for rendering the tables symmetrical when pairs are taken out of any similar
class and tabulated without regard to the relative magnitude of the character in the
two individuals of the pair, i.e., Weatherby's record places the individuals simply in
order of birth and not of darkness or lightness of coatcolour.
Table VII. gives the value of the chief numerical constants deduced from the twenty
four eyecolour tables in Appendix II.*
(11.) On the Mean Eyecolour having regard to Sex and Generation. — In order to
test the degree of weight to be given to our conclusions, I have drawn up a table o*
probable errors for four typical cases — cases by no means selected to give the
lowest possible values. Further, in Table VIII. I have given the probable error in
the position of the median as determined in terms of the grey, bluegreen range by
the modification of Mr. Sheppard's formula (see p. 95). The grey, bluegreen range
of eyecolour is about onefifth of the total observed range, so that the probable error
in the position of the median varies from about "4 to 1 per cent, of that range. This
is not a large error, but, relative to the small variations of value with generation and
.sex, it is sensible, and we must not draw too fine conclusions on the basis of single
inequalities.
Table VI. Table of Probable Errors
in Eyecolour Data.
Relations.
Vx
Vy 1 f '* ; ^*' ^^
1
Mother and Son . . 0253
Maternal Grandmother
and Granddaughter .! 0348
Sister and Sister . .
Maternal Aunt and 0244
Nephew 0230
•0188
•0350
•0244
•0186
•0431
•0767
•0414
•0267
•0276
•0216
•0255
•0256 0283
•0314 0361
•0216 0234
•0250 ^0302
1
* The theoretical formulae by aid of which these constants were determined, have been indicated in the
earlier part of this memoir, and in Part VII. of the present series on Evolution. The actual work of
reduction has been most laborious, but I trust that our results are free from serious error.
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 105
If we examine this table we see that the error in rj amounts to fix)m '02 to '025
when we have upwards of 1000 tabulated eases, but can amount to '035 when we
have as few as 700 to 750 tabulated cases. An examination of the values of rj in
Table VII. shows us that most of our differences with probable errors taken on this
scale are very sensible. A comparison with Table VIII. shows us that the probable
error of the median is always greater than the probable error of 17, and accordingly
the former, being much easier of calculation, may be taken as a maximiun limit. The
probable errors of f , t.c., the ratio of cTar to oy, are more considerable, amounting to about
•04 for our longer series, and even to "077 in the case of the short series of grand
mother and granddaughter, but in this case f actually takes its maximiun value
of 1'291, so that the error is under 6 per cent. ; in the longer series it is under 5 per
cent. Again, we see that in most cases our differences in the ratio of variabilities are
quite sensible. It must be admitted, however, that the ratio of variabilities as based
on the grey bluegreen range of eyecolour is not as satisfactory as that based on the
bay range of coatcolour in horses. In the latter case, onehalf of the horses fall into
the bay range, but only about a quarter of mankind fall into the grey bluegreen
range of eyecolour, and, fiirther, the appreciation of eyecolour seems to me by no
means so satisfactory as that of coatcolour in horses.
I have tried a further series of values for the ratio of the variabilities by measuring
the ranges occupied not only by the tints grey bluegreen, but by the whole range of
tints 3, 4, 5, and 6 of Mr. Galton's classification (see p. 67). Lastly, I have taken
a third method of appreciating the relative variabilities, namely, by using the method
of column and row excesses, E^ and E^, discussed in Part VII. of this series. While
this method has the advantage of using all and not part of the observations to deter
mine the ratio of crx to oy, and so naturally agrees better with the results based on
the four than the one tint ranges, it suffers from the evil that these excesses can only
be found by interpolation methods, which are not very satisfactory when our classes
are, as in this case, so few and so disproportionate. On the whole, this investigation
of relative variability is the least satisfactory part of our eyecolour inquiry, and I
attribute this to two sources : —
(i.) The vagueness in appreciation of eyecolour when no colour scale accompanies
the directions for observation {cf. p. 103, (c) ).
(ii.) A possible deviation from true normality in the factor upon which eyecolour
really depends {cf. p. 80, (ii) 80).
Lastly, we may note that the probable error in the correlation amounts in most
cases to less than '03, rising only somewhat above this value for grandparental
inheritance, where our series are somewhat short — 650 to 750 instead of 1000. Here
again most of the divergences are quite sensible.
Allowing accordingly for the comparative largeness of our probable errors, we
shall do best to base conclusions on the general average of series ; to insist on general
inequalities rather than on exact (juantitative differences, and to eni4)lia8i8e the
VOL. CXCV. — A. P
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106
PKOFESSOR K. PEAESON AND DR. A. LEE ON
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION.
107
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108 PROFESSOR K. PEARSON AND DR. A. LEE ON
general tendency of a series rather than pick out single differences for special
consideration. If we do this we shall still find that remarkable results flow
from our Tables VII. and VIII., most of which seem hitherto to have escaped
attention.
I return now to the special topic of the present section, the mean eyecolour, after this
lengthy — if needful — digression on the probable error of the data given in our tables.
We may, I think, safely draw the following conclusions : —
(a.) Man has a mean eyecolour very substantially lighter than that of woman.
If we compare the mean eyecolour of father with mother, of son with daughter,
of brother with sister, of grandfather with mother, of uncle with aimt, of grandson
with granddaughter, of nephew with niece, we have the same result — man is distinctly
lighter eyed than woman.
(6.) Tlbcre appears to be a secular change taking place in eyecolour^ but this 15
more marked and definite in the man than in the woman.
Thus we have the following mean values for 77 in classes, which must roughly
represent successive generations : —
Grandfather. . . 3658 1 .^aac. Grandmother . . '8757 1 .ggoo
3658 1 .4,49
5241 J
Grandmother .
Mother . . .
. 8757
•8290
5^29 \ 6484
7039 J
Daughter. . .
Granddaughter.
. 7524
•8508
}^
Father
}•
^"^ •^•'"•' )^ 6484 — 8— — . . . '—J. .8016
Grandson. . . ,
Another comparison may be made by noting that grandsons are darker than
grandfathers, sons than fathers, nephews than uncles. Similarly, granddaughters are
lighter than grandmothers, daughters than mothers, but nieces are not lighter than
aunts, as we might have expected. Thus, while the records show a definite darkening
of the eyes of men, there appears to be a certain but less sensible lightening of the
eyes of women. Again, the younger generations are much closer in eyecolour than
the older generations. The difierence in eyecolour between grandsons and grand
daughters, sons and daughters, nephews and nieces is only about 15 per cent, of the
grey bluegreen range, but for fathers and mothers it is 30 per cent., and for grand
fathers and grandmothers 50 per cent.
When we realise that difference in eyecolour appears to be a sexual character, the
true explanation of this secular change in eyecolour becomes stiU more obscure.
If the lighter eyecolour of the older generation be due to an effect of old age, why
is it conspicuous only in the male and not in the female ? Why is the mother sensibly
darker than the daughter, but the father sensibly lighter than the son ?
Further, supposing light eyes much commoner among our grandfathers than among
their grandsons, and dark eyes among our grandmothers than among their grand
daughters, we cannot attribute the great approach in eyecolour to a blending of the
parental characters, for, as we shall see later, eyecolour does not seem to blend, it is
rather an exclusive character. We should, therefore, be thrown back on prepotency
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 109
of the mother — a conclusion possibly warranted by our results in the case of
daughters, but not in the case of sons. Again, why was there such a marked
difference in eyecolour between the men and women of three or four generations
back ?* And if it was a sexual character, why is it disappearing ? Was it not,
perhaps, a racial difference ? Light and dark eyes are not unusually associated with
distinct races, and it is just possible that the change in eyecolour is a product of
reproductive selection ; the old blueeyed element of the population may be dwindling
owing to the greater fertility of the women of darkeyed race, and thus without any
obvious struggle for existence and survival of the fitter, the blueeyed race may be
disappearing from England, as the Langobard type has so largely gone from Italy and
the Frank from France, t It will not do to be dogmatic about these matters, but the
more one measures characters in different generations, the less stable do races appear
to be. We speak of the national characters of the Englishman or the Frenchman
based upon our experience of how these races have acted in past history, but
although there has been no great racial invasion nor struggle, can we safely assert
the physical characters of the Englishman today do not differ substantially from
those of the Englishman of the Commonwealth ? It seems to me that the possibly
continuous change of characters in a race, not subjected to very apparent internal or
external struggle, is a problem of the highest interest to the anthropologist and
.ultimately to the statesman.
Whatever be the explanation of this secular change in eyecolour, it appears to
correspond singularly^ enough to the secular change we have noted in the coatcolour
of thoroughbred horses — in the older generation the sexes differ more widely than in
the yoimger.
(e.) TJie maternal male relative {grandfather^ and uncle) is substantially lightereyed
than the patei^nal male relative {grandfather and uncle). — I see no explanation for this
curious result, but it seems worth while to specially note it, for there are curious
anomalies in the inheritance through the various male and female lines which may
find their complete explanation some day when more and possibly more trustworthy
characters have been investigated.
(12.) On the Variability of Eyecolour with reference to Sex and Class. — The
determination of the relative variability of not exactly measurable characters is, as
we have already seen (p. 105), a somewhat delicate problem. It is more so in the case
of eyecolour in man than of coatcolour in horses, for there is greater difference in
the means, and accordingly the ratio of crx/oy, as found from the ratio of the
" excesses" (p. 105), will be even less reliable.^ The class indices corresponding to the
* Mr. JJalton's records went back to greatgrandfathers, many of whom accordingly appear in our
data for grandfathers.
t See Note II. at the end of this paper.
J The relative variability of all classes was worked out for thoroughbred horses by the " excess "
method, and in only one case — that of dam and colt — did it differ from the bay range method in its
determination of the class with the greater variability.
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110 PROFESSOK K. PEARSON AND DR. A. LEE ON
grey bluegreen range are also not entirely satisfactory in their results, nor those
taken for a still larger range covering tints 3, 4, 5, and 6, or bluegreen, grey, hazel,
light brown, and brown, which cover roughly about 1*5 to 1'6 times the standard
deviation. We shall now consider the results of three methods of considering the
relative variability, (a) jfrom the excesses given in columns 1 and 2 of Table VIII. ;
(fi) from the grey blue green range given in columns 3 and 4 of Table VIII. ; and (y)
from the range of tints 3 to 6 inclusive given in columns 5 and 6 of Table VIII. As
we have already indicated, these methods are not likely to give the same relative
magnitude numerically for the variabilities ; we must content ourselves if they agree
in making the ratio of cr^ to Cy greater or less than unity. Now, in the twentytwo
cases
a and fi disagree in 10 cases.
15 and y disagree in 7 cases.
a and y disagree in 5 cases.
Further, for the five cases in which a and y disagree, those for father and son,
paternal grandfather and grandson, maternal uncle and nephew, show so little
difference of variability in the two sexes that both methods give sensibly the same
results, i.e., equality of variability. In the cases of the paternal grandfather and
grandchildren, the two methods diverge rather markedly.
It will be of interest accordingly to work out the probable errors as given by the
excess method for one, say the first of these cases. The theory is given in Part VIL of
the present series. Here Ei = 275*165, E^ = 309"013, whence we find probable error
of El = 17*273, probable error of Eg = 16*925, correlation between errors in E^ and
Eg = — 4424, probable error in crj/crg = '0394.
Thus the probable error in the ratio of the variabilities is about 4 per cent., and of
the order of the quantities by which we are distinguishing the relative size of Ci
and (Tg.
Further, there is another source of error in evaluating Ei and Eg due to the
method of interpolation used, and this would still further increase the probable error
in cTi/cTg. We cannot therefore lay any very great stress on the manner in which the
ratios of variabilities for the paternal grandfather and grandchildren have swung
round from (a) to (y).
A further examination shows us that in all five cases wherein y differs from a it is
in accord with fi. I shall accordingly take y as the standard criterion, but in those
cases in which it has agreement with a, its conclusions must be given greater weight.
(a.) On the Relative Variability of Sex in Eyecolour. — The following male groups
are more variable than the corresponding female groups : —
Sons of fathers than daughters of fathers.
Sons of mothers than daughters of mothers.
Brothers of brothers than sisters of sisters.
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. Ill
Grandsons (in four series) than granddaughters (in four same series).
Nephews (in four series) than nieces (in four same series).
Fathers (in two series) than mothers (in two series).
Grandfathers (in foiu* series) than grandmothers (in four like series).
Uncles (in four series) than aunts (in four like series).
The following female groups are more variable than the corresponding male
groups : —
Sisters of brothers than brothers of sisters.
Wives than husbands.
We have thus 21 series with male preponderance against only two with female
preponderance of variability.
Again, the mean range of tints 3, 4, 5, 6 in 22 male series equals 1*5424 cr^, and
in 22 female series equals 1*6740 oy, or we have enough evidence to show that the
ratio of male to female variability is about 1*08.*
This greater variability of the male in eyecolour is of considerable interest. It
does not appear to be a result of sexual selection, for so far as our comparatively small
series weighs, husbands are less variable than wives. That mothers are, however, less
variable than fathers seems to indicate that darkeyed women are more fertilet than
lighteyed, for we must bear in mind that mothers have on the average a darker eye
colour than wives. We have thus again reached the same conclusion as before,
namely, that a darkeyed element in the population with a prepotent fertility is
replacing the blueeyed element.
The other female exception to the general rule of greater variability in the eye
colour of the male is that in mixed families the sisters appear to be more variable
than their brothers, notwithstanding that brothers of brothers are more variable than
sisters of sisters. In other words, so far as eyecoloiu* is concerned an exceptional
man is more likely to have brothers than sisters, but an exceptional woman also is
more likely to have brothers than sisters. The inference is not very strong, as the
excess method (a) makes brothers of sisters and sisters of brothers of sensibly
equal variability; it rests therefore on (fi) and (y) only. Still it deserves ftdler
investigation.
(6.) Let A and B be two grades of relationship, of which A refers to the older
generation, and A and B refer to either sex. Then the variability of all the A's
* It is worth noting that the ratio of male to female variability in the coatcolour of horses is r05 (see
p. 96). In both cases the female is darker, i.e., has less of " colour " ; thus if we could take a coefficient of
variation ratio instead of standard deviation ratio as the test, we should find the difference of variability
less, possibly even zero.
t For if mothers are to ]ye less variable than wives, their distribution must be more compressed round
the mean than that of wives ; this denotes that fertility is correlated with eyecolour, and the darker eye
colour goes with the greatei fertility. [See Note II. at end of memoir, however.]
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112 PROFESSOR K, PEARSON AND DR. A. LEE ON
who have female B's is invariably greater than the variability of all the A's who
have male B's.
The law appears to be universal, at least it is absolutely true for all the 10 cases to
which we can apply it. Thus the father of sons is less variable than the fether of
daughters, the maternal grandmother of grandsons less variable than the maternal
grandmother of granddaughters, or the paternal uncle of nephews less variable than
the paternal uncle of nieces. In other words, although women appear, in eyecolour,
to be less variable than men, they spring from more variable stocks.
This law is a remarkable one, but in face of the evidence for it, it seems difficult to
doubt its validity. Should it be true for more characters in man than eyecolour,*
the conclusions to be drawn from it will be somewhat farreaching, however difficult
it may be to interpret its physiological significance.
(c.) On the Relative Variability of Different Generations. — We have already had
occasion to refer to the general rule that the older generation will be found less
variable than the younger, for it is in itself a selection, namely, of those able to
survive and reproduce themselves. But this rule is obscured in the present case by
several extraneoxis factors, thus : —
(i.) The male is sensibly more variable than the female, consequently it is quite
possible that an elder male generation should appear more variable than a younger
female generation.
(ii.) There appears to be a secular change in eyecolour going on. Thus while the
grandparental population is a selection from the general population, the general
population, at a given time, is a selection from that of an earlier period.
Thus, taking means in the cases of the grandparental and avuncular relationships,
we have from (y) the following results : —
The father is more variable than son and than daughter.
The mother is less variable than son and more than daughter.
The grandfather is more variable than grandson and than granddaughter.
The grandmother is less variable than grandson and more than granddaughter.
The uncle is more variable than nephew and more than niece.
The aunt is less variable than nephew and more than niece.
In other words, the older generation is always more variable than the yoimger,
except when rule (a), that the male is more variable than the female, comes in to
overturn this law. If we confine ourselves to comparisons of the same sex the rule is
seen to be universal.
We are thus forced again to ask for an explanation of the decreasing variability of
eye colour, and can only seek it in that secular change we have several times had
* Fathers of daughters are more variable in stature than fathers of sons (*Phil. Trans.,' A, vol. 187,
p. 274). I propose to reinvestigate the question with regard to mothers from the material of my family
measurement cards, which is far more extensive than the material I had at my disposal in 1895.
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 113
occasion to refer to. Mean and standard deviation of eyecolour appear to have
changed sensibly during the few generations covered by Mr. Galton's eye data.
It is difficult to understand how any obsciu^ity about the recording of eyecolours
could lead to anything but chaos in the numerical results. It does not seem to
me possible that such results as we have reached under (a), (6), and (c), namely,
greater variability in the male, greater variability in the stock of the female, and
secular change in variability, can be due to any process of recording. I am forced to
the conclusion that they are peculiar to the character under investigation, and are
not due to the manner of taking the record or of dealing with it arithmetically. I
have purposely avoided drawing attention to small diflPerences and forming any con
clusions which did not depend on whole series of groups and substantial averages.
1(13.) On the Inheritance of Eyecolour, (a.) Assortative Mating. — Before we
enter on the problem of inheritance, it is as well to look at the substantial
correlation obtained between the eyecolour in husband and wife. When in 1895
I reached the value '0931 ± '0473 for stature, I wrote, " we are justified in con
sidering that there is a definite amount of assortative mating with regard to height
going on in the middle classes."* Since then we have worked out the coefficients of
correlation in stature, forearm, and span for 1000 husbands and wives (instead of
200) fiom my family datat cards. The results, which are very substantial, will be
dealt with in another paper, and amply confirm my view that assortative mating is
very real in the case of mankind. The value ('0931) cited above is in close agree
ment with the result now reached ('1002 db '0378) for eyecolour in the same
materiaL The correlation between husband and wife for two very divergent
characters is thus shown to be about "1, or is 25 per cent, greater than is required
between first c(msins\ by the law of ancestral heredity.
This remarkable degree of likeness between husband and wife — ^the scientific
demonstration that like seeks like — cannot be overlooked. It shows that sexual
selection, at least as far as assortative mating is concerned, is a real factor
of evolution, and that we must follow Darwin rather than Wallace in this
matter. § *
(6.) Collateral Heredity. First Degree. — I deal first with this form of heredity,
as it offers least points for discussion. The values of the correlation '5169 for
brothers, and '4463 for sisters and sisters are considerably less than what we have
found for coatcolour in horses, but, like the value '4615 for brothers and sisters, are
substantially greater than '4 to be expected from the immodified Galtonian law.
They could be reached by making y greater than unity in my statement of the law
of ancestral heredity.  They could also be given by the law of exclusive inheritance
* * Phil. Trans.,' A, vol. 187, p. 273.
t See also * Grammar of Science,' second edition, pp. 429437.
X ' Roy. Soc. Proc.,' vol. 62, p. 410. § « Roy. Soc. Proc.,' vol. 66, p. 140 ei seqS
II I have considered possible explanations of thia apparently large assortative mating (i.) in both stature
VOL. CXCV, — A. 9
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114 PROFESSOR K. PEARSON AND DR. A. LEE ON
(see p. 90) with a certain degree of prepotency in the individual pairing. As we
have already noted, collateral inheritance of the first degree alone considered
will not enable us to discriminate between blended and exclusive inheritance.
We note that the male in collateral inheritance predominates over the female,
brothers being more alike than sisters in eyecolour, and brother and sister more
alike than sister and sister. The mean value for inheritance in the same sex is,
however, greater than the value for inheritance between opposite sexes {cf, p. 102).
(c.) Collateral Heredity. Second Degree, — A very cursory inspection of the
coefiicients of correlation for the eight series of avuncular relationships shows us
that it is quite impossible that the mean value should be '15 as required by the
Galtonian Law. The average value of the avuncular correlation is '2650, and of
the regression of nephew and niece on uncle or aunt is '2733. The probable error
of the former result will not be more than '02, and of the latter something greater,
as the ratio of the variabilities is open to larger error. This mean value is accord
ingly, within the limits of errors of investigation, identical with the '25 to be expected
on the theory of exclusive inheritance. It is a value which appears to be quite
impossible on the theory of blended inheritance even with my generalised form of
the ancestral law.
We may draw several other important conclusions from our table of avuncular
correlations :—
(i.) Nephews are more closely related to both uncles and aunts than nieces are.
This is true in each individual case, whether it be judged by correlation or regression.
The mean correlations for uncles and for aunts are as '3081 to '2219 respectively.
(ii.) Uncles are more closely related to nephews and nieces than aunts are. This
is true for three out of the four individual cases ; in the fourth case the difference
is of the order of the probable error of the difference. The mean correlations ot
nephews and nieces are as '2923 and "2377 respectively.
(iii.) Paternal uncles and aunts are more closely correlated with their nieces and
nephews than maternal uncles and aunts. The mean values are as '2719 to '2580.
(iv.) Resemblance between individuals of the same sex is closer than between
individuals of opposite sex. The mean values for the avuncular correlation in the
same sex and in the opposite sex are respectively "2751 and •2549.
(v.) Uncles are more closely related to nephews than aunts to nieces (mean
correlations as '3455 to '2046). In fact, generally, we see a very considerable
preponderance of heredity in the male line so far as these avuncular relations for
and eyecolour, being characters of local races, or even families, and the husband seeking his wife in his
own locality or kin; (ii.) in a possible coiTelation of homogamy and fertility. See *Eoy. Soc. Proc,,*
vol. 66, p. 28. Neither seem very satisfactory. Consciously or unconsciously, man and woman appear to
select their own type in eyecolour and stature, until they are apparently more alike than such close blood
relations as first cousins ! Until we know how far this correlation extends to other characters, it would,
perhaps, be idle tp draw conclusions as to its bearing on widely current views as to first cousin marriage».
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 115
eyecolour extend. It is noteworthy that while the two highest correlations are
reached for nephew with paternal and with maternal uncles, nearly the two lowest
are found for niece with paternal and with maternal aimts. Without laying
special stress on each small diflTerence, it must be admitted that the avuncular
correlations vary in a remarkable manner with sex, and differ very widely from
the practical equality of resemblance which we might d prioii have expected to
exist in this relationship.
(d.) Direct Heredity. First Degree. — Here we have a mean value of the paternal
correlation = '4947. This is in excellent agreement with the '5 to be expected by
our theory of exclusive inheritance ; it is thus in practical agreement with the value
of the parental correlation obtained for the inheritance of coatcolour in horses. It
would not be inconsistent with a high value for y in the theory of blended inheri
tance, but such a value of y is rendered impossible by the values we have obtained
for collateral heredity (see 'Roy. Soc. Proc.,' vol. 66, p. 140 et seq.).
We may draw the following special conclusions: — (i.) The son inherits more
strongly from his parents than the daughter, the mean correlations are as '5160
to "4733 ; (ii.) The son inherits more strongly from his father than his mother,
and the daughter more strongly from her mother than her father.
This is part of the general principle which we have seen to hold, namely : that
change of sex weakens the intensity of heredity.
The correlation of father and daughter appears to be abnormally below the other
three, but something of the same kind has been noted in certain stature data ; as it
is, the high correlation of father and son renders the mean paternal correlation with
o&pring ('4936) sensibly equal to the mean maternal correlation ('4956).
(e.) Direct Heredity. Second Degree. — If we take the mean value of the eight
grandparental correlations, we find it equals '3164, while the meaji value of the
regression of offepring on their grandparents is '3136. These results are absolutely
incompatible with the '15 required by Mr. Galton's unmodified theory, and they in
fact put the theory of blended inheritance entirely out of court. At the same time,
unlike the cases of parental, avuncular, and fraternal inheritance, they cannot be said
to be in good agreement with the value '25 required by the theory of exclusive
inheritance. We have to admit that our grandparental data are shorter series than
in the other cases, and that guesses as to grandparents' eyecolour, based on memory,
miniatures, &c., were more likely to be made. Further, such guesses might easily
be biased by a knowledge of the eyecolour of more recent members of the family.
Still a reduction from '32 to '25 is a very large reduction, and we have to remember
that for long series in the case of the thoroughbred horses, with no such guessing at
colour as may occur with ancestors' eyes, we found '3353 for the maternal grand
sires, a result in excellent agreement with the '3343 found for the maternal grand
fathers in the present case. Thus while the theory of exclusive inheritance without
reversion suffices to describe the quantitative values we have found for the parental,
g 2
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116 PROFESSOR K. PEARSON AND DR. A. LEE ON
the avuncular and the fraternal correlation in the cases of both horse and man, it is
yet in both these cases unsatisfactory so far as the grandparental inheritance is con
cerned. It may be imagined that if we allowed for reversion, we might emphasise
the grandparental correlation beyond the value '25 suggested by theory. But I
have shown in my memoir on the " Law of Reversion," that with the parental correla
tion as high as '5, we cannot hope to have the grandparental correlation even with
reversion higher than •25. (See * Roy. Soc. Proc.,' vol. 66, p. 140 et seq.) Clearly the
values obtained for grandparental correlation in this paper — the first I believe
hitherto investigated — seem to present anomalies which oiu* theory of blended
inheritance totally fails to accoimt for, and which may require some modification of
our views on reversion before we can meet them on our theory of exclusive
inheritance.
I note the following general results deduced from our values of the grandparental
correlations : —
(i.) Grandsons are more closely correlated with both grandparents than grand
daughters are. This is true for three out of the four cases ; the exception, maternal
grandmother, is covered by another rule (iv.). The mean correlation for grandparents
and grandsons is '3294, and for grandparents and granddaughters '3039.
(iL) Grandfathers are more closely correlated with grandchildren than grand
mothers are. This is true in three out of the four cases, the fourth being again
subject to rule (iv.). The mean correlations for grandfathers and grandmothers are
'3675 and "2658 respectively.
(iii.) Paternal grandparents appear to be more closely correlated with their
grandchildren than maternal grandparents, the average values of the two correlations
being '3236 and '3097 respectively.
(iv.) Resemblance between individuals of the same sex is closer than between
individuals of the opposite sex. The mean values for the grandparental and
grandchild correlation in the same and the opposite sexes are '3329 and '3004
respectively.
(v.) Grandfathers are more closely related to grandsons than grandmothers to
granddaughters, the mean correlations being as '3965 and '2693 respectively! It
will be noted at once that these five rules are identical with those we have obtained
for the avuncular correlations. So that there is small doubt that they are general
rules relating to all grades of relationship for this character.
It seems to me probable that the correct form of (iii.) is : Paternal grandfathers are
more highly correlated with grandchildren ('4006) than maternal grandfathers ('3343),
and paternal grandmothers (*2468)less highly correlated than maternal grandmothers
(•2851). I have not stated the rule in this form, because it is not confirmed by
the corresponding results for uncles and aunts. Paternal uncles ('3024) are more
closely correlated with nephews and nieces than maternal uncles (*2722), but paternal
aunts ('2414) are slightly more instead of less correlated with nephews and nieces
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MATHEMATICAL CONTEIBUTIONS TO THE THEORY OF EVOLUTION. 117
than maternal aunts ('2338). I consider, however, that the correlation of paternal
aunt and nephew ('2837) in our series is abnormally high.
Now it will, I believe, be seen that the investigation of the eight avuncular and
the eight grandparental relationships, here made for the first time,* enables us to
draw far wider conclusions than when, as hitherto, only parental and fraternal corre
lations are dealt with. In making the subjoined general statements, however, I
must emphasise the following limitations : —
(a.) The rules are deduced only from data for one character in one type of life.
(fi.) This character appears to be undergoing a secular change, a change very
possibly due to a correlation between eyecolour and fertility in wcwnaan. Thus such
a change might not unlikely differentiate the male and female influences in heredity.
My conclusions, definitely true for eyecolour in man, and at the very least
suggestive for investigations on other characters in other types of life, are : —
(L) That the younger generation takes, as a whole, more after its male than its
female ascendants and collaterals.
(ii.) That the younger generation is more highly correlated with an ascendant or
collateral of the same than of the opposite sex.
(iii.) That the younger generation is more highly correlated with an ascendant or
higher collateral reached by a line passing through one sex only than if the line
changes sex.
Thus correlation is greater with a paternal uncle than with a maternal uncle, or
with a maternal grandmother than a paternal grandmother.
(iv.) Males are more highly correlated with their ascendants and collaterals than
females are.
The above rules apply to the averages ; individual exceptions will be generally
found to arise fi:om a conflict of rules. Thus (ii.) and (iii.) may in special cases come
into conflict with (i.). When we have more data for a greater variety of characters,
we shall see better the relative weight of these rules in cases where they conflict.
[f.) Exclusive InheHtance in EyeColour. — A cursory examination of the eye
colour records shows at once how rare is a blend of the parental tints. Even when
such is recorded, it is by no means clear that we have not to deal with a medium tint
which is really a case of reversion to a medium tinted ancestor. The failure of eye
colour to blend is, I think, well illustrated by what Mr. Galton has termed cases of
" particulate " inheritance. In the thousands of eyecolours I have been through, I
noticed some halfdozen cases only in which the two eyes of the same individual were
of different tint, or the iris of one pupil had streaks of a second tint upon it.t
* I anticipate equally valuable results when characters are first correlated for the nine possible cousin
series.
t In the same manner the occurrence of particulate inheritance in coatcolour in horses may be really
an argument against the existence of blends. In the many volumes of the studbooks I have examined,
the recorded instances of pieljalds are vanishingly few in niunber.
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118 PROFESSOR K. PEARSON AND DR A. LEE ON
If we allow that it is from the theory of exclusive inheritance that we must seek
results in the present cases, we see that for parental, collateral, and avuncular relation
ships we get quite excellent results, but that the grandparental relationship is some
what anomalous. A priori it might appear that reversion would aid us in increasing
the correlation between offspring and remote ascendants. But, as I have shown else
where,* this superficial view of reversion forgets that the parents as well as the
offspring revert, and if we increase the grandparental correlation above '25, we at once
reach difficulties in the values of the parental correlation, provided we adopt what
appear to be reasonable assumptions as to reversion being a continuous and decreasing
factor from stage to stage of ancestry. I am inclined accordingly to suspend judg
ment on the grandparental relationships, thinking that the smallness of the number
of families dealt with in Mr. Galton's data (200) may have something to do with my
peculiar results. Meanwhile I shall endeavour to get the remaining six grandparental
tables for thoroughbred horses worked out, and see whether they confirm the high
values ah'eady found for the two maternal grandsires and oflfepring, or give an average
value much nearer '25.
That the reader may see at a glance the general results hitherto obtained in this
and other papers, I append the following table of inheritance : —
* See my paper on "The Law of Reversion/' 'Roy. Soc. Proc./ vol. 66, p. 140 et scq. Also *The
Grammar of Science/ second edition, 1900, pp. 48696, " On Exclusive Inheritance."
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 119
Table IX. — Theoretical and Actual Results for Inheritance.
Relationship.
Theory.
Man.
Horse.
Hound.
Daphnia.
Blended
inherit
ance.'
Exclusive
inherit
ance.*
Statiu^e*.
Head
index.*
Eye
colour.*
Goat
colour.^
Coat
colour."
Spine."
Parental ....
Midparental . . .
Grandparental . .
G. Grandparental .
Avuncular . . .
Whole sibling . .
Half sibling . . .
•3000
•4242
•1500
•0750
•1500
•4000
•2000
•5000
•2500
•1250
•2500
•4 to 10
•2 to 5
•3355
•4745
•4034
•3348
•4735
•4025
•4947
•3166
•2650
•4749
•5216
•3353
•6329
•3550
•3507
•1340
•0404
•5170
•1646
[•3295]
•4660
[•1360]
•6934
* Mr. Galton's unmodified hypothesis. See " Law of Ancestral Heredity," * Roy. Soc. Proc./ vol. 62
p. 397.
2 Without any reversion. See "Law of Reversion," * Roy. Soc. Proc.,' vol. 66, p. 140 et seq. The values
for the fraternal correlation depend on the degree of prepotency of either parent within the union.
8 See * Phil. Trans.,' A, vol, 187, p. 270.
^ See 'Roy. Soc. Proc.,' vol. 62, p. 413. The paternal correlations, for reasons stated in the paper, are
excluded from the result.
^ See p. lis et seq. of the present memoir.
^ See p. 98 et seq. of the present memoir. The grandparental correlation is based on two cases only.
7 See *Roy. Soc. Proc.,' vol. 66, p. 140 et seq.
^ Sec * Roy. Soc. Proc.,* vol. 65, p. 154. I have deduced the value for parents and grandparents from
Dr. Warren's results for midparent and midgrandparent. The value for whole siblings I obtained from
Dr. Warren's measurements, which he with great kindness placed at my disposal.
(14.) Conclusions. — The course of this investigation has not been without diffi
culties, and I am fully prepared to admit that more obscurity and greater probable
errors are likely to arise when we deal with the inheritance of a character not directly
measurable, than when we take that of a character to which we can at once apply a
quantitative scale. But I contend that many of the characters, the inheritance of
which it is most important to investigate, do not at present, and perhaps never will,
admit of a quantitative measurement. We can arrange in order, we can classify, we
can say more or less intense, but we cannot read off value on a scale. It is just such
characters also, which the not highly trained observer can most easily appreciate and
record. Hence we have been compelled to devise some method of dealing with them,
and the present paper illustrates how the method invented can be applied to reach
results of considerable interest and of substantial validity.
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120 PROFESSOR K. PEARSON AND DR. A. LEE ON
In order to illustrate the method, I chose two characters, coatcolour in horses and
eyecolour in man, which seemed sufficiently diverse both as to origin and species.*
The new method enabled me to reach results for halfbrethren, grandparents and
uncles and aunts, which had not yet been independently considered. The conclu
sions arrived at for eyecolour in man at no point conflict with those for coatcolour
in horses, and both in the main accord with the theory of exclusive inheritance with
out reversion herein developed. We find —
(i.) No approach to a single value for the coefficient of inheritance for each grade
of relationship; it varies widely with the sex, and the line through which the
relationship is traced.
(ii.) No approach in average values to those which would be indicated by Mr.
Galton's Law.
Nor does the modification of Mr. Galton's Law, which I have termed the Law of
Ancestral Heredity, give better results. For, if we cause it to give the parental
values, it then renders results inconsistent with the fraternal values.
(iii.) There is agreement with the theory of exclusive inheritance without reversion
for the parental, avuncular and fraternal series ; but there is some anomaly in the
case of grandparental inheritance. This requires further investigation, and possibly
a modification of our views on the nature of reversion.
We want a list formed of characters in various types of life, which are supposed to
be exclusively inherited, and then experiments ought to be made and statistics col
lected with regard to these characters. It is in this field of exclusive inheritance
that we must look for real light on the problem of reversion.
If we consider the three known forms of inheritance, the blended, the exclusive,
and the particulate (which may possibly be combined in one individual, if we deal
with different organs) ; if we consider further that these forms may possibly have to
be supplemented by others not yet recognised {e.g.y reversional theories depending,
say, on heterogamous unions), then it would appear that the time is hardly ripe even
for provisional mechanical theories of heredity. What we require to know first is,
the class of organs and the types of life for which one or other form of inheritance
predominates. As variation in no wise depends on the existence of two germplasms,
so biparental heredity can by no means be treated as the result of their simple quanti
tative mixture ; the component parts of these germplasms corresponding to special
characters and organs, must be able to act upon each other in a variety of qualita
tively different ways. To adopt for a moment the language of Darwin's theory of
pangenesis, the multiplying gemmules from an organ in the father must (i.) cross with
gemmules from that organ in the mother, and the hybrid gemmules give rise to
blended inheritance, (ii.) must without crossing multiply alongside the gemmules of
the mother, and give rise to particulate inheritance, (iii.) must alone survive, or alone
* Since supplemented by my investigations, based on Mr. Galton's data, for coatcolour in hpunds,
* Roy. Soc. Proc.,' vol. 66, p. 140 et seq.
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 121
be destroyed in a struggle for existence with those of the mother, and give rise to
exclusive inheritance. And all these three processes may be going on within the
same germplasm mixture at the same time ! Even without using the language of
gemmules, processes analogous to the above must be supposed to take place. Thus a
quantitative " mixture of germplasms " becomes a mere name, screening a whole
range of mechanical processes ; and very possibly a new one could be found for each
new form of heredity as it occurs. Such processes like the old ones would still
remain without demonstrable reality under the veil of " mixture of germplasms."
What I venture to think we require at present is not a hypothetical plasmic
mechanics, but careful classifications of inheritance for the several grades of rela
tionship, for a great variety of characters, and for many types of life. This will
require not only the formation of records and extensive breeding experiments, but
ultimately statistics and most laborious arithmetic. Till we know what class o^
characters blend, and what class of characters is mutually exclusive, we have not
within our cognizance the veriest outlines of the phenomena which the inventors of
plasmic mechanisms are in such haste to account for. Such inventors are like planetary
theorists rushing to prescribe a law of attraction for planets, the very orbital forms of
which they have not first ascertained and described. Without the observations of
Tycho Bbahe, followed by the arithmetic of Kepler, no Newton had been possible.
The numerical laws for the intensity of inheritance must first be discovered from wide
observation before plasmic mechanics can be anything but the purest hypothetical
speculation.
Appendix I.
Tables of Colour Inheritance in Thoroughbred Racehorses, extracted by Mr. Leslie
BramleyMoore jfrom Weatherby's Studbooks.
Table of Colours.
1 = black (bl.) 9 = chestnut or bay (cli./b.).
2 = black or brown (bl./br.). 10 = chestnut (ch.).
3 = brown or black (br./bL). 11 = chestnut or roan (ch./ro.).
4 = brown (br.). 12 = roan or chestnut (ro./ch.).
5 = brown or bay (br./b.). 13 = roan (ro.).
6 = bay or brown (b./br.). 14 = roan or grey (ro./gr.).
7 = bay (b.). 15 = grey or roan (gr./ro.)
8 = bay or chestnut (b./ch.). 16 = grey (gr.).
VOL. CXOV. — A. R
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 137
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VOL CXCV. — A.
Digitized by VjOOQ IC
138
PROFESSOR K. PEARSON AND DR. A. LEE ON
Appendix II.
Tables of Eyecolour Inheritance in Man, extracted by Kabl Pearson from
Mr. Francis Galton's Family Records.
1 = light blue.
2 = blue, dark blue.
3 = bluegreen, grey.
4 = dark grey, hazel.
Table op Tints.
5 = light brown.
6 = brown.
7 = dark brown.
8 = very dark brown, black.
This grouping is not quite in keeping with more recent divisions of eyecolour, but
being that adopted by Mr, Galton in his original collection of data, it could not be
modified in accordance with present practice.
Tables for the Direct Inheritance of Eyecolour. First Generation.
I. — Fathers and Sons. 1000 Cases.
Failiers.
GQ
Tint.
1.
2.
3,
4. i
5.
6.
7.
8.
Totals.
1
9
12
5
5 ,
1
2
34
2
10
163
65
36 '
1
7
15
4
301
3
10
73
124
41
1
12
18
6
284
4
4
21
34
55
11
11
1
137
5
1
2
2
1
5
6
1
26
12
19 '
1
19
16
6
100
7
1
23
16
14 ,
11
31
2
98
8
1
4
8
10
1
7
10
41
Totals
36
322
264
180 ;
i
5
64
101
28
1000
II. — Fathers and Daughters. 1000 Cases.
Faihers,
Tint
1.
2.
3.
4.
5.
6.
7.
8.
Totals.
1
4
9
11
4
1
2
1
32
'n
2
11
139
57
31
6
24
6
273
^
3
9
73
111
38
1
15
19
3
269
1
4
5
43
34
54
2
10
14
3
165
5
1
3
3
1
8
M
6
1
45
13
19
23
15
3
11?
7
2
27
10
12
7
41
6
105
8
8
4
2
11
4
29
Totals
32
345
243
158
3
67
127
25
1000
Digitized by
Google
MATHEMATICAL CONTRIBUTIONS TO THE THEOEY OP EVOLUTION.
139
in. — Mothers and Sona 1000 Casea
Mothers.
Tint.
1.
2.
3.
4.
5.
6.
7.
8.
Totals.
1
5
14
6
3
1
6
35
2
12
119
83
29
8
20
21
9
301
3
13
54
113
35
4
37
14
8
278
4
3
21
26
54
1
17
6
6
134
5
1
1
3
5
6
1
9
26
10
1
30
24
3
104.
7
1
9
19
16
18
31
7
101
8
7
15
4
1
3
5
7
42
Totals
35
234
289
151
15
129
101
46
1000
IV. — Mothers and Daughters. 1000 Cases.
Mothers,
Tints.
1.
2.
3.
4.
5.
6.
7.
■ 8.
Totals.
1
5
15
3
2
2
2
2
31
2
7
99
67
29
2
15
23
13
255
3
7
77
111
38
1
26
14
6
280
4
5
22
34
46
2
27
21
7
164
5
2
2
3
1
2
1
11
6
13
27
20
1
35
17
7
120
7
13
21
16
1
19
26
9
105
8
1
5
7
2
1
4
12
2
34
Totals.
25
246
272
153
13
129
117
45
1000
Tables for the Collateral Inheritance of Eyecolour.
V*.— Brothers and Brothers. 1500 Cases.
First Brother,
1
First.
1
1.
2.
3.
4.
5.
6.
7
8.
Totals.
1
8
2
3
4
19
1
2
36
202
23
17
6
4
3
291
3
16
182
209
26
4
2
2
441
1
4
6
36
71
84
7
2
206
5
3
2
1
1
7
f
6
3
56
50
39
34
5
6
193
fc
7
6
37
76
48
1
36
36
2
242
8
4
24
26
18
8
6
15
101
Totals
79
542
460
237
1
96
55
30
1500
T 2
Digitized by
Google
140
PROFESSOR K. PEARSON AND DR. A. LEE ON
OQ
'^
VI*. — Sisters and Sisters. 1500" Cases.
First Sister.
First.
1.
2.
3.
4.
5.
6.
7.
1
8.
Totals.
1
10
2
1
1
14
2
17
147
29
6
10
6
2
217
3
10
136
186
24
9
5
3
373
i
3
75
94
66
1
10
249
5
3
5
2
2
1
13
6
2
57
69
55
5
62
9
2
251
7
4
56
61
52
10
59
49
291
8
2
20
10
7
6
13
26
8
92
Totals
48
496
455
213
23
144
106
15
1500
V^ — Brothers and Brothers. Symmetrical System.
Tint.
1.
2.
3.
4.
5.
6.
7.
8.
Totals.
1
16
38
19
10
3
6
6
98
2
38
404
205
53
3
62
41
27
833
3
19
205
418
97
2
64
78
28
901
4
10.
63
97
168
1
46
50
18
443
6
3
2
1
1
1
8
6
3
62
54
46
1
68
41
14
289
7
6
41
78
50
1
41
72
8
297
8
6
27
28
18
14
8
30
131
Totals
98
833
901
443
8
289
297
131
3000
VI^ — Sisters and Sisters. Symmetrical System.
Tint.
1
1.
2.
3.
4.
5.
6.
7.
8.
Totals.
1
20
19
11
4
2
4
2
62 •
2
19
294
165
81
3
67
62
22
713
3
11
165
372
118
5
78
66
13
828
4
4
81
118
132
2
56
62
7
462
5 !
3
5
2
4
6
11
6
36
6
2
67
78
56
5
104
68
15
396
7
4
62
66
62
11
68
98
26
397
8
2
22
13
7
6
15
26
16
107
Totals
1
62
713
828
462
36
395
397
107
3000
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION.
141
<S
VII, — Brothers and Sisters. 1500 Cases.
Brcihers.
Tint.
1.
2.
3.
4.
5.
6.
7.
8.
Totals.
1
5
9
18
4
1
37
2
20
163
101
36
28
19
13
380
*5
3
9
98
193
50
37
17
14
418
■S
4
5
36
49
67
3
28
13
16
217
il ' 5
2
5
1
1
2
2
3
16
6
3
47
41
27
4
42
17
14 1
195
7
4
34
49
22
3
27
30
19 !
178
8
1
1
10
7
1
10
8
22 ■
59
; Totals
47
399
463
208
11
174
107
,1 1
1
1500
Table for Assortative Mating in Eyecolour.
Vni. — Husbands and Wives, 774 Cases.
Husbands,
Tint
1.
'■
3.
4.
1
5.
6.
7.
8.
Totals.
1
2
13
4
3
1
2
25
2
6
87
42
26
16
13
6
196
1
3
6
56
93
31
1
16
11
6
220
4
4
32
35
18 1
1
15
6
1
1 112
5
6
1 1
1
7
6
1 2
38
27
10 !
1
12
10
1
101
7
1 5
20
28
7
1
6
12
4
83
8
! 2
8
8
2
2
4
4
30
Totals
: 27
1
254
242
98
4
68
59
22
774
Tables for the Direct Inheritance of Eyecolour. Second Generation.
IX. — Paternal Grandfather and Grandson. 765 Cases.
Paternal Gramdfather,
First.
i
! 1.
2.
3.
4.
5.
6.
7.
8.
Totals.
1
4
10
3
1
3
21
2
7
115
31
20
1
6
13
3
196
3
5
64
109
21
10
22
4
235
4
2
25
40
21
9
13
6
116
6
1
1
2
6
14
32
11
15
5
2
79
7
4
16
16
9
1
11
21
2
80
8
6
5
3
16
6
36
Totals
i 22
250
236
83
•
2
66
93
23
765
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142
PBOFBSSOE K. PEAESON AND DR. A. LEE ON
X. — Paternal Grandfather and Granddaughter. 681 Cases.
Paternal Grandfather,
First.
1.
2.
3.
4.
5.
6.
7. •
8.
Totals.
1
3
6
4
5
1
1 '
20
2
2
94
32
10
2
6
16
4
166
1
3
5
67
71
17
9
20
3
192
4 1
4
36
33
26
1
10
9
3
121
5
3
4
1
1
2
1 1
12
'^
6
1
16
21
11
15
6
4 '
74
1
7
3
10
20
11
1
8
15
3
71
8
i
2
5
1
1
1
10
5
25
Totals
18
233
190
82
5
51
79
23
681
Table XI. — Maternal Grandfather and Grandson. 771 Cases.
Maternal Grandfather.
Tint. 1
i
1.
2.
3.
4.
5.
6.
7.
8.
Totals.
1
3
11
3
1
1
19
2
8
113
46
22
1
13
5
3
1 211
3
10
87
89
12
11
8
3
220
^ 1
5 !
6
4
33
1
25
35
1
25
22
—
15
6
2
117
2
84
2
7
14
7
4
7
1
22
26
6
2
9
10
4
80
8
—
12
12
6
—
4
3
1
38
Totals
1
28
304
237
76
3
66
40
17
771
I'
I
Table XII, — Maternal Grandfather and Granddaughter. 687 Cases.
Maiemal Grandfaiher,
Tint.
1.
1
2.
3.
4.
5.
6.
7.
8.
Totals.
1
1
1
3
7
2
1
14
2
8
84
35
11
13
6
2
159
3
11
67
76
18
7
15
5
199
4
7
41
40
14
15
11
5
133
6
—
5
2
2
1
—
10
6
4
21
32
1
16
5
2
81
7
4
15
14
8
1
7
19
2 1
70
8
1
5
5
1
1
5
4 1
21
Totals
35
241
211
55
1
61
63
20 !
687
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION.
143
Table XIII. — Paternal Grandfather and Grandson. 741 Cases.
Paternal Grandmother.
1
Tint.
1.
2.
3.
4.
5.
6.
7.
8.
Totak.
1
1
2
7
1
1
2
3
17
2
6
62
69
22
4
15
25
4
207
3
4
31
95
22
1
25
33
9
220
4
3
18
36
20
4
15
16
4
116
5
—
—
1
1
1
3
6
1
10
23
6
2
16
10
4
72
7
3
15
10
4
1
14
13
11
71
8
1
10
3
3
—
«
5
8
36
Totals
19
148
243
79
13
92
106
41
741
I"
I
Table XIV. — Paternal Grandmother and Granddaughter. 717 Cases.
Paternal Grandmother.
Tint.
1.
2.
3.
4.
5.
6.
7.
8.
Totals.
1
3
3
2
6
2
16
2
7
53
56
14
2
13
28
8
181
3
8
35
65
22
6
29
28
7
200
4
4
29
36
20
3
16
23
8
139
5
1
—
3
1
5
1
—
11
6
9
29
2
4
27
8
3
82
7
—
10
15
12
2
10
12
7
68
8
1
2
1
2
1
 4
4
5
20
Totals
21
141
208
75
18
104
110
40
717
I
Table XV. — Maternal Grandmother and Grandson. 756 Cases.
Maiemai OrcMdmotker.
Tint.
1.
2.
3.
4.
5.
j
6.
7.
8.
Totals.
1
1
10
1
3
1
3
1
19
2
10
68
53
23
24
13
13
204
3
9
39
67
38
32
23
11
219
4
5
6
3
34
19
1
11
30
1
10
—
19
8
4
117
2
84
^
20
z
24
18
1
7
2
9
23
11
1
17
17
—
80
8
—
4
6
5
—
'< 6
'
7
31
Totals
1 25
t
184
181
121
1
122
85
37
756
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144
PROFESSOK K. PEARSON AND DR. A. LEE ON
Table XVI. — Maternal Grandmother and Granddaughter. 739 Cases.
Maternal Grandmother.
Tint
1 1.
2.
3.
4.
6.
6.
7.
8.
Totals.
,
1
2
16
18
;§
2
7
66
34
13
—
21
15
6
162
§*
3
12
62
55
25
1
27
23
5
210
^
4
6
32
36
25
23
15
7
144
i
5
1
3
2
3
3
12
1
6
1
14
21
11
27
17
2
93
7
1 "■"
19
17
7
—
16
17
3
76
8
1
5
4
2
—
7
3
3
24
Totals
28
212
170
85
1
124
93
26
739
Tables for the Collateral Inheritance of Eyecolour. Second Degree.
XVIL — Paternal Uncle and Nephew. 1290 Cases.
Paternal Uncle,
Tint.
1
1.
2.
3.
4.
5.
6.
7.
8.
Total.
1
4
10
11
6
1
4
5
2
43
2
11
136
98
40
26
48
12
371
§
3
8
84
157
26
1
27
54
7
364
^
4
29
69
36
1
19
27
12
193
5
2
1
2
1
6
6
1
31
35
7
1
30
19
3
127
7
2
21
27
24
1
13
34
11
133
8
11
7
6
10
8
11
53
Total
26
324
405
145
5
131
196
58
1290
XVIII. — Paternal Uncle and Niece. 1128 Cases.
Paternal Uncle.
Tint.
1.
1
2.
3.
4.
5.
6.
7.
8.
Total.
1
2
10
6
2
1
6
2
29
2
7
85
61
27
29
26
13
248
^^
3
1 6
82
126
29
1
26
43
7
319
•^
4
2
47
73
40
1
29
40
5
237
e;
5
1
8
1
4
1
5
4
24
6
1
26
35
12
1
8
42
3
128
7
: 1
20
26
19
22
26
7
120
8
1
4
2
3
5
3
6
23
Total
1 26
282
329
136
3
121
191
47
1128
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION.
145
Table XIX. — Maternal Uncle and Nephew. 1242 Cases.
Maternal Unele.
Tint.
1.
2.
3.
4.
5.
6.
7
8.
Totals.
1
1
8
13
3
3
4
1
33
2
17
137
71
29
19
14
9
296
g
3
10
128
153
26
29
34
3
383
1"
4
2
50
62
28
22
14
1
179 I
5
1
1
2
6
4
33
29
12
35
20
3
136
7
1
33
40
11
26
27
2
140
8
9
17
23
8
3
13
73
Totals
35
399
385
132
142
117
32
1242 i
Table XX. — Maternal Uncle and Niece. 1434 Cases.
Maternal Uncle.
Tint.
1.
2.
3.
4.
5.
6.
7.
8.
Totals.
1
2
15
9
2
2
1
31
2
6
99
76
23
18
13
11
246
3
18
100
108
23
37
36
10
332
4
4
72
64
28 
16
21
9
214
5
14
2
3
8
5
32
6
5
38
41
10
23
11
4
132
7
1
27
25
7
19
14
3
96
8
15
5
6
9
11
6
51
Totals
36
380
330
102
132
112
42
1134
*
Table XXI. — Paternal Aunt and Nephew. 1186 Cases.
Paternal Awnt.
Tints.
1.
2.
3.
4.
5.
6.
7.
8.
Totals.
1
6
13
5
3
1
4
6
4
42
2
19
113
83
45
36
29
5
330
3
10
81
147
30
—
29
35
8
340
4
8
28
66
38
—
18
22
11
191
5
.i__
__
._»,
_
—
—
^
6
3
23
35
12
1
35
10
5
121
7
5
22
28
19
18
16
5
112
8
1
4
9
8
—
6
4
13
47
Totals
52
284
373
155
2
148
121
51
1186
VOL. CXCV, — A.
V
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146
PBOFESSOR K. PEAESON AND DR A. LEE ON
Table XXII. — Paternal Aunt and Niece. 1149 Cases.
Paternal Aunt.
Tints.
1.
2.
3.
4.
5.
6.
7.
8.
Totals.
1
2
11
2
3
_
2
11
2
33
2
16
89
62
37
2
25
40
14
284
3
12
93
119
40
3
41
26
12
346
4
10
36
62
43
5
25
21
11
213
6
5
7
—
—
1
3
—
16
6
1
24
33
16
1
29
19
5
128
7
2
20
28
12
2
10
22
4
100
8
—
7
—
4
—
5
9
4
29
Totals
42
28.5
313
155
13
138
151
62
1149
t
Table XXIII. — Maternal Aunt and Nephew. 1145 Cases.
Maternal Aunt.
First.
1.
2.
3.
4.
5.
6.
7.
8.
Totals.
1
4
8
7
3
2
3
1
28
2
5
117
81
29
43
29
6
310
3
1
73
132
38
57
43
3
347
4
1
20
54
27
1
21
11
—
135
5
3
2
—
1
—
—
6
6
—
24
35
22
—
30
23
3
137
7
—
26
29
20
1
26
25
8
136
8
—
14
6
10
—
12
4
2
47
Totals
11
282
346
151
2
192
138
23
1145
Table XXIV. — Maternal Aunt and Niece. 1058 Cases.
Maternal Aunt,
First.
1.
2.
3.
4.
5.
6.
7.
8.
1
Totals.
1
2
3
10
15
1
6
87
86
31
—
23
14
12
; 258
3
3
71
125
32
1
49
41
3
1 326
4
—
39
51
31
1
33
19
6
, 180
5
—
4
6
1
—
8
5
3
! 27
6
1
25
47
10
—
24
9
2
' 118
7
—
30
29
11
14
18
10
112
8
—
5
4
5
—
2
4
3
; 23
Totals
11
264
358
121
2
163
no
39
1058
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION.
147
[Notes added July 3, 1900.
Note I. Inheritance of Temper and Artistic Instinct. — In additional to the fraternal
correlations given on p. 102, T have dealt with Mr. Galton's statistics for the inheri
tance of good and bad temper given in his 'Natural Inheritance' (p. 235). The
following gives the distribution of good and bad temper among 1,294 brethren, as
deduced by Mr. Yule.
First Brother.
I
I
Co
Good Temper.
Bad Temper.
Totals.
Good
temper.
330
255
585
Bad
temper.
255
454
709
Totals
585
709
1294
The correlation is '3167.
A like table is that for artistic instinct in the direct line : —
Parentage.
Artistic.
Nonartistic.
Totals.
1
296
173
469
1038
"1
372
666
Totals
668
839
1507
In this case the correlation is '4039.
The fraternal correlation is somewhat low. The exact significance of the parental
correlation is also somewhat vague, as the parentage is classified as artistic when
one or both parents are artistic. But the two tables are very suggestive, they
indicate how the new method will enable us to deal quantitatively even with
characters like temper and artistic instinct to which it is impossible to apply directly
a quantitative scale. With the introduction of a third or medium class, I believe it
will be possible to obtain excellent results for heredity from very simple observations,
and I have in hand at the present time a large series of observations on collateral
heredity based upon such simple classifications. The reader should further consult
u 2
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148
PROFESSOR K. PEARSON AND DR. A. LEE ON
Mr. G. U. Yule's remarks on the association of temper and of artistic instinct in his
memoir on '* Association/' ' Phil. Trans./ A, vol. 194, p. 290, 1900.
Note IL On the Correlation of Fertility and EyeColour. — In the course of the
present paper I have frequently referred to a probable influence of reproductive
selection as the source of the progressive change in eyecolour, i.e., to a possibility that
eyecolour is correlated with fertility. I saw from Mr. Galton s tables that in
many cases the whole family had not been recorded, probably the eyecolour of the
dead or of absentees being unknown. It appeared to me accordingly that it would
be impossible to deal directly with the problem of fertility. However, it has since
occurred to me that there is nothing likely to give the missing members of families a
bias towards one rather than another eyecolour, and that we may simply treat them
as a purely random subtraction from the total results. Assuming this, Mr. L. N.
FiLON, M.A., has prepared for me tables of father's and mother s eyecolour and of
the recorded number of their children. From these* I take first the following
results, premising (i.) that I call " light eyed," persons with eyecolours 1, 2 and 3, and
" dark eyed," persons with eyecolours 4, 5, 6, 7, 8, i.e., drawing the line between light
and dark grey ; (ii.) that I take as small families those with 0, 1, 2, or 3, recorded
children and us large those with 4 or more recorded children.
Father.
Light Eyed.
Dark Eyed.
Totals.
1
313
Ul
139
280
1
454
403
1
264
577
Totals
857
Mother
Light Eyed.
Dark Eyed.
Totals.
^
M
253
202
455
&
3
225
169
394
Totals
478
371
849
* Correlation tables were prepared of the size of families to 15, and of the eyecolours 1 to 8, but it
does not seem needful to print them in extenso.
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MATHEMATICAL CONTMBUTIONS TO THE THEORY OF EVOLUTION. 149
We have, accordingly, by the method of the present memoir : —
Correlation of size of family with darkness of eyecolour
= '0595, for fathers.
= — 0239, for mothers.
The former is just sensible, the latter hardly sensible relative to the probable error.
So far as they can be relied upon, they would denote that fathers have more children
the darker eyed they are, and mothers more children the lighter eyed they are.
This is in accordance with the result given in the memoir, that the modern
generation is darker than its male and lighter than its female ancestry, but it is not
the explanation given in the text, although it is probably the true one. If it be the
true one, dark fathers and light mothers ought to present the most fertile unions,
and it seemed desirable to test this directly. We have already seen that there
exists an assortative mating in eyecolour, like tending to mate with like, the
CO efficient of correlation being about '1 ; hence if we were to correlate the eyecolour
of mothers and fathers, i.e., husbands and wives weighted with their fertility, we
ought to find this result substantially reduced. The following is the table : —
Fatliers.
I
d
Light Eyed.
Dark Eyed.
Totals.
<D
>^
P^
1183
612
1795
^
bO
>A
nj
a>
,>»
W
826
455
1281
^
u
ee
Q
1
Totals
2009
1067
3076
We find ?• = '0239, or the correlation has been reduced to a fifth of its previous
value, and is now of the order of its probable error. To mark still further this
increased fertility of heterogamous unions, I add two further tables, giving the mean
number of recorded oflfepring for various classifications of parental eyecolour.
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150 MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION.
Fathers. Fathers.
I
Colours
13.
Colours
48.
Average
of totals.
if
3"
383
457
407
§ 00
382
373
379
Average
of totals
383
417
394
^
Colours
12.
Colours
38.
Average
of totals
386
3"
319
452
P
398
396
397
Average
of totals
368
409
394
The first table entirely confirms all the conclusions reached, — dark fathers and
light mothers are most fertile absolutely and in union. The second table shows
that it is the bluegreen and grey rather than pure blueeyed mothers who are most
fertile. This supplementary investigation accordingly seems to support the view of
the text of the memoir, namely that reproductive selection is the source of the
secular change in eyecolour noted, only the prepotent fertility which is replacing the
blueeyed element is in the first place that of the darkeyed male, and only in the
second place due to mothers having eyecolours dark or light other than true blue.
We seem accordingly in eyecolour to have reproductive selection working through
heterogamy rather than through homogamy as in the case of stature.* The effect,
however, is like, — the progressive elimination of one type of character.]
♦ See * Roy. Soc. Proc.,' vol. 66, p. 30, and vol. 66, p. 316 et seq.
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[ 151 ]
IV. 071 Simultaneous Partial Differential Equations.
By A. C. Dixon, Sc.D.
Communicated by J. W. L. Glaisher, Sc.D.
Eeceived May 9— Read June 15, 1899.
Contents.
Pages
§ 1. Introductory 151
§§ 2 — 9. On " bidifferentials," or the elements of double integrals, and on the conditions to be
satisfied in order that a given bidifferential expression may be a complete bidiffer
ential 152—159
§§ 10—13. Theory of equations linear in the Jacobians of two unknown functions; their solution
reduced to the formation of complete bidifferentials 159 — 162
^ 14 — 30. Theory of other simultaneous partial differential equations in two independent and
two dependent variables. A method of solution, with examples of its application.
One pair of variables is said to be a " bifunction " of other pairs when its bidifferen
tial can be linearly expressed in terms of theirs : this idea is of importance in con
nection with the derivation of all possible solutions when complete primitives are
known. Construction of bif unctions in some cases 162—181
^31 — 42. Differential equations of the second order with one dependent and two independent
variables. A method of sohition, with examples 181 — 191
§ 1. In this paper, without touching on the question of the existence of integrals
of systems of simultaneous partial diflferential equations, I have given a method by
which the problem of finding their complete primitives may be attacked.
The cases discussed are two : that of a pair of equations of the first order in two
dependent and two independent variables, and that of a single equation of the second
order, with one dependent and two independent variables.
I follow, as far as possible, the analogy of the method of Lagrange and Charpit,
and with this object introduce the conception of the " bidifierential " or differential
element of the second order, which bears the same relation to a Jacobian taken with
respect to two independent variables as a differential does to a differential coefficient.
The solutions considered are, in general, complete primitives, that is, such as contain
arbitrary constants in such number that the result of their elimination is the system
of equations proposed for solution. The existence of such primitives is sufficiently
established (see the papers of Frau von Kowalevsky and Professor Konigsberger,
quoted hereafler) ; it will therefore be assumed, and the object of the investigation
5.11.1900
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152 MR. A. C. DIXON ON SIMULTANEOUS
will be to find conditions that must be satisfied by the equations of the solution and
to put these conditions in a convenient form for solution by inspection.
I should add that I am greatly indebted to the referees for their suggestions and
for help in removing obscurities.
To the list of authorities given by Dr. Forsyth (' Theory of Differential Equations/
Part I., pp. 299, 331), may be added the following : —
Julius Konig. Math. Annalen, vol. 23, pp. 520, 521.
Leo Konigsberger. Crelle, vol. 109, pp. 261340.* Math. Annalen, vol. 41,
pp. 260285.t Math. Annalen, vol. 44, pp. 1740.
Ed. v. Weber. Mlinchen Ber., vol. 25, 423442.
J. M'CowAN. Edinb. Math. Soc. Proc, vol. 10, 6370.
Hamburger. Crelle, vol. 110, pp. 158176.
C. BouRLET. Annales de I'ficole Normale (3), vol. 8.
RiQUiER. Comptes Rendus, vols. 114, 116, 119. Annales de Tficole Normale
(3), vol. 10.
Lloyd Tanner. Proc. Lond. Math. Soc, vols. 711.
J. Brill. Quarterly Journal of Math., vol. 30, pp. 221242.
Several of the above papers are only known to me through abstracts.
On Bidifferentials,
§ 2. The idea of a ** complete differential" plays an important part in the theory
of differential equations. In this paper I shall try to show the importance of an
extension of the same idea to differential elements of higher orders, such as enter
into multiple integrals.
An expression Xc/x + Ydy is called a complete differential when X, Y are functions
of the independent variables x, y, such that
8Y/ax = ax/ay.
If this is the case, then, under certain restrictions, the value of J(Xc/a: + Yo??/) depends
only on the limiting values of the variables, and not on the intermediate ones by
which these limits are connected, or, as generally expressed, on the path along which
the integral is taken.
This depends on the theorem that
fdY 8X\
\{Xdx + Ydy) = \\(^i^yxdy
* For reasons stated below, I am not in agreement with the results given in the latter part of this
paper.
t In this paper it should be noticed that the equations (52) on p. 266 are not more general than (46).
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PARTIAL DIFFERENTIAL EQUATIONS. 153
when the single integral is taken round the boundary of the area over which the
double integral is to extend.
Further, X, Y are in this case the partial derivatives of a single function.
§ 3. Let us consider the double integral
\\(Xdydz + Ydzdx + Zdxdy),
where X, Y, Z are functions of the independent variables or, y, z. It is known
that this, taken over a closed surface under certain restrictions, is equal to the triple
integral
J J J(aX/aa; + a Y/ay + dZ/dz) dx dy dz
taken over the space enclosed by that surface.
Hence, if dX/dx + SY/dy + dZ/dz = identically the double integral taken over
a closed surface vanishes, and taken over two open surfaces with the same boundary
has the same value ; that is to say, the value of the double integral depends on the
values of a;, y, z at the boundary only, and not, under certain restrictions, on the
form of the surface enclosed by the boimdary.
By analogy we may call the element of the double integral a " complete double
diflTerential," or a " complete bidiflferential " \mder these circumstances ; the condition
that X dy dz + Y dzdx + Zdx dy may be a complete bidiflferential is thus
dX/dx + dY/dy + dZ/dz = 0.
§ 4. A complete bidiflferential may be expressed as a single term, such as dti dv.
For let u^ vhe two independent solutions of the equation
SO that n = a, v = 6 are integrals of the system
cte/X = dy/Y = d2/Z;
then ' X = ^^, Y = ^^, Z = ^^>,
3(y,«)' o(z,x)' d(x,y'
6 being some multiplier,
d ax 8Y az_ 3(g,tt,t>)
3lK dy dz d{x, y, z)'
Since the last vanishes identically ^ is a function of u, v only ; a function w of tt, v
may be foimd, such that Zwjdu = 6, and thus
X
VOL. CXCV. — A.
__ 9(w, v) y _ 9(w, v) „ _ d(w, v)
d(y,z)' '~8(2,r)' ^a(r.y)
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154
MR. A. C. DIXON ON SIMULTANEOUS
Now in finding the value of the double integral taken over a part of any surface,
it will be natural to suppose the coordinates of any point of such a surfia.ce to be
fiinctions of two parameters, say p, q, and to transform the integral into one taken
with respect to these. The integral as transformed is
\\{
9(y.2)
+ Y
3(g, x), ,j^(x,y)
+ Z
3(f2)^ a(2>,j)^ d(p,
nW"^
and the known values in terms of p, q are to be substituted for x, y, z and their
derivatives,
ITie subject of integration is
dw ^t dio dy dw dz hv dx dv dy^ dvdz
3r 9p 8y 9p 92; 9p ' 9iB 2jp dy dp dz dp
dw dx dw dy^ dw dz Sv^dx dv dy dv ^
dx dq dy dq dz dq* dxbqdydqdzdq
or
The integral is therefore
d(Wt v)
and if we take a single element we may write
Xdydz + Ydzdx + Zdxdy=^ dw cZv,
dropping the parameters jp, q^ since the values which x, y, z have in terms of them
are immaterial.
This equation is meaningless unless the expression in terms of parameters is under
stood. The same is true of ordinary differentials. If when w is a function of sc, y, z
we write
da^^dx + ^dy + ^dz,
we mean that if x, y, z are supposed to be any functions whatever of a single
parameter jp, then
die du dx du dy ^^du^ dz^
dp '^ dx dp dy dp dz dp'
This equation being true quite independently of the expressions assumed for x,y,zm
terms of jp, we drop the denominator dp for convenience ; but in modem works on the
Differential Calculus it is quite understood that a differential by itself is meaningless
apai't fi'om this or some equivalent convention.
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PARTIAL DIFFERENTIAL EQUATIONS. 155
§ 5. The fiinctions w, v are not uniquely determined. They may be replaced by
W, V, where W, V are functions of w, v, one of which, say W, is arbitrary, while V is
only restricted by the condition
d(w, v)
The transformations of ?/;, v which are allowable will thus form a group. For a
single integral the operations of the corresponding group consist in the addition of
different constants, that is, in varying the constant of integration ; the theory of
periodic functions is connected with discontinuous subgroups of this. It is possible
that an investigation of the discontinuous subgroups of the group of transformations
of two variables which leaves their bidifferential unchanged may lead to an extended
theory of periodic functions of the two variables.
§ 6. The finding of the functions w, v may be considered as the indefinite integration
of the bidifferential expression. It is simplified by Jacobi's theory of the last
multiplier, which is here a constant.
we have Xdy — Ydx = ;^ ^^^ "" ^ ^^ >
and thus, on the supposition that v is constant,
, XdvYdx YdzZdy ZdxXdz
dw = — ^ = 5 ^ = 5
ov cv ov
dz dx xy
_ (fi Z  vY)dx + (i/X ^ \Z)dy + (XY  /xX)rfg
Hence w may be found, if v is known, by integrating this last expression on the
supposition that v is constant ; X, /x, i/ may have any values and the constant of
integration is to be replaced by an arbitrary function of v. Thus, when one of the
functions w^ v is known, the other is found by ordinary integration. The only
restriction on the one found first is the equation
§ 7. Let us now suppose a greater number of independent variables. Let ?^ be a
function of aj^, Xg . . . o?^.
We have the relation
X 2
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156 MR. A. C. DIXON ON SIMULTANEOUS
Here the ditterentials represent simultaneous infinitesimal increments, those of the
independent variables being arbitrary. The equation may also be interpreted by
supposing «!, X2 . . . x„ to depend in any manner on a single parameter p, when the
equation
du * 3m dxr
^'P ""r=i3'V dp
holds whatever functions of the parameter we suppose x^ . . . aj^ to be.
To get the idea of a double differential we must suppose two sets of simultaneous
infinitesimal increments ; denote them by d, 8. The bidifferential of a;, y is then
dx . 8y — 8x . dy* This vanishes if a?, y are not functionally independent, just as dx
vanishes if re is a constant. The analogy is very clearly shown if we say that dx
vanishes when some function i^{x) vanishes, dx dy vanishes when some function
^(Xy y) vanishes.
If Uy V are functions of n independent variables x^^ x^ . . . aj», we have
dw = S 5 dXr, 8w = 2 5 SaJr
** ov * 3t?
di; = S ^ dXry 8t; = S ^ 80:^, and hence
du.hv — Zu.do — % S ^ ^ {dXr . hXf — dx^ . 8a:^),
or
dudv = X ^ ' dxr dx.
the summation being taken over all pairs of different suffixes r, s. Hence the
expression for du dv is formed by multiplying together
2 5— dXr and 2 ^ cte;.
with the conventions
dxdy = — dy dx,
dx dx = 0.
We shall often use the notation d{x, y) for dx dy.
§ 8. For the purpose of double integration of such an expression as 2 X„ d{xr, a?,),
in which the coefficients X are ftinctions of acj . . . a:„, it is natural to suppose a:^ , . . a;»
expressed throughout the range of the integration in terms of two parameters, say
jp, q. The integral thus becomes
* The dot is used here and throughout the paragraph to distinguish multiplication in the ordinary
algebraic sense from multiplication according to the Grassmann conventions stated at the end of the
paragraph.
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PAETIAL DIFFERENTIAL EQUATIONS. 157
w
If X„ = ^ * \ for all pairs of suffixes, the subject of integration in the last
integral is d{tf^ v)/d{pf q)^ so that the. integral becomes Hdudv. Its value will
therefore only depend on the values of w, v, that is of jc,, a;^ . . . x^, at the boundary
of the range of integration, and not on the form of the relations giving 0:^, x^ . . .
in terms of ^, q^ which define the particular siu'face over which the integral is taken.
In this case we may write
2 Kr, d{Xry X,) = d(u, v)
and call it a complete bidifferential. It is easily seen that the coefficients X satisfy
the relations
X,,X(,. + XhX,, + X^.X« = Oi ...... . (1)
.' t+t+t" f^''
for aU combinations of suffixes, where it is understood that the term Xr,c?(a:^, x,) may
be also written X^d{x,^ Xr), so that
Xf , = — X,.,.
The conditions (2) are those which must be satisfied in order that the value of the
double integral may depend only on the boundary. The difierence of two values
of the double integral, for which the same boundary is assumed, will be its value over
a closed surface passing through the boundary curve, and this may be transformed
into the triple integral
fffi,(t + l: + th^^^
taken through the voluriie of any solid bounded by this closed surface. Hence this
integral must vanish for any solid. By taking an infinitesimal solid, for every point
of which all but a;,, Xr, Xg are constant, we find the condition (2).
The conditions (2) would be satisfied by an expression which was the sum of two
or more complete bidifferentials, but (1) in general would not.
§ 9. We next try to find whether these c6nditions are sufficient as well as
necessary. Now all the coefficients X cannot vanish. Suppose that X^g does not,
then we have firom (1)
and in virtue of these all the conditions (1) are satisfied.
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158 MR. A. C. DIXON ON SIMULTANEOUS
Taking the values thus given for Xr„ X,>, X« we have
~ x,jl aPi . ■*" eb,/ ■•■ XijV &•.• ■•■ ar,/ "^ x„V a^, "^ a^. /
Y
/^ J. ^^ J. ^'Y^ J ?^\ 4 ^?t/^' U ^^
x„ axjg Xfr axjg Xjj axjg
X]2 a^'i Xjg aa7j Xj3 atv
= 1^ (IW) + ^ (r2i) + 1^' (2st) + I" (sit) + y («2r) + ^ (1^)
^13 *^12 ^12 ^12 ^12 ^12
M2*^2
+ 1^ (21«) + ^ (21r) + 1^ (21t).
Ajj Aj, a.12
Thus the conditions (2) are not independent, but all follow from those in which at
least one of the suffixes 1, 2 enters. If they are satisfied then the equations
H n
S Xi^ dXr == 0, 2 Xgy dXr =
rss2 r=l
can be satisfied by two integrals of the form w = a, v = h; that is, these last
equations will give Xj, ic^ as functions of the rest, such that
&i Xg^ ^ Xrt
For the conditions necessary and sufficient* for this are the vanishing of such
expressions as
J^_ ?!l 4. ?«? . ^_ ?!1J I ^1 ^ ^1 _ 9 Xrt ^ Xe^ j^ Xrt ^ Xyi 9 Xrt
d^, Xjg Xjj &i Xi2 Xjg OTg Xjj 9.ZV Xj2 Xj3 Oit'i Xij Xjj ocj X^
in which 1, 2 may be interchanged and i% s are any two of the other suffixes. This
expression may be written
i;, <^l*) " §; (^2s) +  (12r) + ^ ^ (Xi2 X„ + X,, X^ + X^ X,.),
SO that it vanishes and the conditions of integrability of the equations XXj^Xr = 0,
r
l,X2rdXr = are satisfied. If w = a, v = 6 are the integrals, then, since tK^dxr
r
does not contain the diflferential of x^, we must have
* For proof of this statement see Fobsyth, * Theory,* part I., pp. 4361.
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PARTIAL DIFFERENTIAL EQUATIONS. 159
d(u, v)
and Xir =
3(«i, av)'
In like manner Xgr = 6 ^7 — :, the multiplier being the same.
Hence ■ ^ = *^!
Since this vanishes for all combinations of suffixes, ^ is a fimction of u, v, and if
another function of them, ti;, is so chosen that
we shall have t X^, d{Xr^ x,) = d{Uy v) = d{w, v).
r,«
Linear Differential Equations.
§ 10. If t^ = a is an integral of the linear partial differential equation
where X^, X^ . . . X^^.! are ftmctions of x^y . . . x^^^i^ then n satisfies the condition
S X,^ = 0,
r=l OXr
and the complete differential du is a linear combination''^ of the determinants
dxiy dx^i dx^ • • . dx^y dXf^+i
^l9 ^> ^ • • • ^J ^+1
the coefficients in the combination being usually functions of x^, . . . Xm^i.
If u = a is a conunon solution of the above equation and of
^1 ai^i + • • • +^ Stew ^+^'
then, in like manner, duiaa, linear combination of the determinants
* This is generally expressed by saying that *' it = a is an integral of the equations
Xi X2 Xn+i
For the sake of the analogy with the work of § 11, I prefer the phrase in the text, which expresses no
more and no less than the one generally used.
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160 ME. A. C. DIXON ON SIMULTANEOUS
dxu ^^2> • • • r^n+u
^l> ^> • • • J Xn+i,
X' Y ' Y'
but in general, of course, it will not be possible to combine them so as to form a
perfect differential.
§ 11. An analogous process of integration may be given for two simultaneous
equations
2{Aj,(My  Pjqi)} + SB.?>. + tCqi + E = 01
t{A:^{piqjpjqi)} + tB\'Pi + tC',qi + E'^0} * * * * (3)'
in which the coefficients A, B, C, E, A', B', C, E' axe functions of n independent
variables, ajj, x^ . . . x^y and two dependent y, 2, and
To fix the ideas, take n = 3 and let x^ x^ stand for y, z respectively, A;4 for C„
Aj5 for — Bj, A45 for E, and make similar changes in the accented letters. Then, if
tc =z a^ V = h are two equations constituting a solution,* a, h being arbitrary con
stants, we must have
IJ = 1,2..8
'd(a.bXj)
(4);
and the values thus given for p^^ g^, j?2> Q'sj i^8> Q's ^aust satisfy the equations (3) identi
cally, since a, & are supposed arbitrary. The equations to be solved are thus reduced
to others which are linear and homogeneous in the Jacobians, and which do not
contain the dependent variables.
The equations (4) give two of the Jacobians of a, v linearly in terms of the
others ; if we substitute tor these two in the identity
'*(''.'')=a^)'^^A
we find that d{u, v) is a linear combination of the determinants of the matrix of tent
columns.
* This solution will not be a complete primitive unless a certain number of other arbitrary constants are
involved as well as a, 6, a supposition which is neither made nor excluded.
It may be well to point out that the solution here assumed consists of two equations, and not of one
equation involving an arbitrary function ; in fact, any solution whatever necessarily consists of two
equations, and one point of the present method is that these are to be sought together, not successively.
t For n independent variables the number of columns in the matrix will be (n + 1) (n + 2), the
number of rows being still thiee.
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PARTIAL DIFFERENTIAL EQUATIONS. 161
d{x^, x^), d{x^, a,), . . . d{xi, Xj) . . , d{x^, x^)
A' A' A' A'
There are thus eight bidiflferential expressions, and the problem is to be solved by
finding such multiples of these as, when added together, will form a complete
bidifferential.
§ 12. As in the case of Lagrange's linear equation, this will generally, in practice,
be done by inspection, and the method will be useful for finding solutions in finite
terms — when such exist. But in any case,* whether the inspection is successful or
not, there can be no doubt of the existence of suitable multipliers, in infinite number.
For it is certain that the equations (3) have — possibly among other solutions — an
infinity of solutions, each involving two arbitrary constants at least, and any one of
these may be written t^ = a, v = 6, where a, h are the two constants ; m, v are
functions of the variables, but may, of course, be implicit functions of great com
plexity. The functions w, v must satisfy the conditions (4), and it immediately
follows that d{u^ v) must be a linear combination of the determinants of the matrix
formed from (4) as above ; so that a corresponding system of multipliers must exist.
If the solution is not in finite terms it is not likely to be found by inspection, and
it is quite probable that the best way to find it would be by solving the original
equations (3) in series. By whatever means the solution is found, the corresponding
system of multipliers is thereby determined.
If nine solutions of the form w = a, v = 6 have been found, the nine
bidifierentials rf(wi, Vj), ^(ttj* Vg) . . . d{ug^ Vg) must satisfy identically a linear rela
tion, since they are all linear combinations of eight expressions only.
We shall say that one of the nine pairs of functions is a " bifunction " of the other
eight pairs.
The following is, then, the definition of a bifunction. When the bidiflferentials of
any nimiber of pairs of quantities are connected by an identical linear relation, with
constant or variable coefficients, any one of these pairs is said to be a bifunction of
the rest.
The word bifunction is simply used as an abbreviation — at least for the present. I
am not without hope that at a future time it may be found to have some connotation.
* If one of the dependent variables with its derivatives is altogether absent from the equations (3), or
if it can be made to disappear by a change of the other dependent variable, the equations (3) will in
general have no solution. This case will then be excluded ; it is the only case in which the method of
solution in series (as given, for instance, by Frau von Kowalevsky, * Crelle/ vol. 80) camiot be used
to prove that solutions actually exist.
Another case that may fairly be excluded is that in which all the derivatives of one of the dependent
variables do not occur or may be made to disappear by a change of the other. Such a system is equiva
lent to a single partial differential equation with one dependent variable, since the one whose derivatives
are absent may be eliminated.
VOL. CXCV. — A. Y
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162 MR. A. C. DIXON ON SIMULTANEOUS
It is, of course, evident that if % v are functions of variables a^i, x^ . . . then the
pair u, V 18 a. bifunction of all the pairs that can be formed from ajj, x^ , . . Other
examples will be found later on in the paper.
§ 13. Sometimes solutions exist for systems of partial differential equations in
which the number of dependent variables is less than the number of equations.
If, for instance, with the system just considered we take a third equation of the
same form, the coefficients being distinguished by two dashes, there may be solutions
common to the three equations. If u — a^ v = h give such a solution, then it
follows in like manner that d{u, v) is a linear combination of the determinants of the
following matrix : —
d{x^,x,) . . .
d{xi, xj) .
Ai2, . . .
A,y, . .
A'
^ 12> • • •
A'..
A''
A.% : .
Similarly for a greater number of equations.
Application to other Differential Equations.
§ 14. There are two classes of equations whose solution depends on that of a pair
of such linear homogeneous equations as we have just been considering ; they are,
firstly, systems of two equations in two dependent and two independent variables,
and, secondly, equations of the second order with one dependent variable and two
independent. We shall consider them in order.
Firstly, let y, zhe the dependent variables and .Tj, a;., the independent ; sometimes
we shall write x^ for y and x^ for z. Let Pi^ p^ be the partial derivatives of y and
g^i, q^ those of a;, and let the equations be
/IC^I. ^2. Vy ^y Pu Pzy ?1> Qz) = 0,
Ai^v ^2. y> 2;, ^i, ^2, g'l, q^) = 0.
A complete primitive will consist of two equations connecting x^y x^, y, z and
involving four arbitrary constants. By differentiation these equations yield four
more involving p^, p^y g'l, g'2 ^^ t^® ^^^o equations are supposed to be a complete
primitive it must be possible to find expressions for the four arbitrary constants in
terms of aj^, x^, y, 2, Pi, 5^1, P2> qz I t^® elimination of the four constants must give
/i = 0,/2=0.
Let «!, ag, ag, a^ be the constants, and ^i, t^o, w,, u^ the expressions for them in
terms of x^, Xc^y y, 2, p^, q^y p^^, q.^. Suppose /g, f^yf^.f^ to stand for u^y u^y Vg, %i^
respectively. Then by differentiation we have for any value of the suffix i from
I to 6,
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PARTIAL DIFFERENTIAL EQUATIONS. 163
the letter d being used to denote differentiation with respect to x^ or Xo on the sup
position that the other is constant, while 3 indicates strictly partial differentiation.
Since dpjdxx = dpjdx^, dqjdx^ = dqjdx^^ we find by eliminating the deriva
tives of jpi, gi, ^2, g^2, that
J(^i>i>i. 9^1. 5^2) +i>iJ(y.i>i> <iu 92) + qi^^y Ply qu ^2) + J(^2>P2> 9i> 92)
+ Pr^iVy V^y 9\y 9%) + 92^{^yV2y 9\y 9%) = 0,
and %!, ji, ^1, p^) + ^1%, ^1, p^, p^) + q^J{z, q^, p^, p^) + J{x^, q^, p^, p^)
+ P2^yy 92y Ply Pi) + 92^{^y ?2>Pl>i^2) =
where J( ) denotes the Jacobian of any four of the functions /j, f^yfzyf^^fhyA ^*^^^
respect to the variables specified in the bracket. Of these equations there are thirty,
but since they are given by the elimination of six quantities from twelve equations
only six of the thirty can be independent.
§ 15. One pair of these auxiliary equations will contain Jacobians of /i, /g, ^, f^y
and will in fact express the conditions that the equations
dy = p^dxi + p^dxc^
dz = q^dxi + q^dxc^
shall be integrable without restriction when p^, p,^, q^, q^ have the values given by
the equations /i = =fc^, /g = a^,f^ = a^.
Thus, if a pair of functions ^, J^ can be found satisfying these two auxiliary
equations, the solution can be completed by solving a pair of simultaneous ordinary
equations. (See Mayer's method, Forsyth, * Theory of Differential Equations,' pp.
5962.)
The two auxiliary equations that^^ f^ must satisfy are linear and homogeneous in
their Jacobians, the coefficients of the Jacobians not involving the functions f^, f^ ;
the number of independent variables is apparently eight, but it may be taken as six,
since two of the eight variables x^, Xg, y, z, P\y p^y 9\y 9^ ^^^ given as functions of the
other six by the relations /j = 0,^ = 0, and may be supposed eliminated from^j,/^,
if that is desirable.
The colimins of the matrix formed as at § 11 are the rows of the following array : —
c?(xi, Xg), 0, 0,
d{x^y y\ 0, 0,
d{x^y z\ 0, 0,
d{xzy y)y 0, 0,
(5) d{xz, z), 0, 0,
Y 2
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164 MR. A. C. DIXON ON SIMULTANEOUS
%,«), 0, 0,
^{^u Pi). (91. 93} » {Pi,qi}>
d{x^,p{), 0, {Pi,q'i],
%. Pi)>Pi{qv %} pAps, 9i} + PiiPi, qi),
(10) d{z, Pi),gi{?i, q,], qAPi, qi] + q^iPi, q^],
(^i^!lyPi)>{q^yqi}> {q2,pi],
^iy'Pi\Pi{quqi]> PiiquPi) +Pi{qiyPi]*
d{^>P2)> qdqu qi}y qi{q\>Pi) + qi{qi>P))>
d{yi,qi),pi{qci,Pi] •\P2{qi,Pi}, PifPi.i'al.
c^(2.?i). ^ifo'z.i'i} + qi{qi>Pi}, qi{px.Pi}>
d{xi,qi),{px,qi}y 0, .
(20) d{Xi, q^), (pa, q^}, f i>i, P2} ,
%. 92), I'll;?!, qi) + P2(P2. g'l}* i'2{pi.i>2}.
«^{«. 72), g'lfPl. fZi) + 92(l>2. 9l}. 92{2'.i>2}.
^0^1. M 0. {a^i, 9i} +i>i{y. 9i} + 9i{«.9i} + {a'2. 92} +Pi{y> 92} + ?s{2. 92}.
c?(Pi. 9i). {^\,qi\ +Pi{y, 92} + Q'll^. q%}> {^i>Pi} pAy^Pi)  g'j2.i'2}>
(25) ci(pi,q.2),  {Xi.^ri} Pi{y,9i}9i{2.9i}. i^i^Pa) PziV^Pi)  92KP2}.
%2. 9i). W 92} + P2(y.9a} + qd^^qi), {aJi.Pi} +2>i{y.Pi} + gJ^^Pil.
^(P2.</2).{a;2.!?i}P2{y.'?i}?2{2.g'i}. {3^2,^1} +i?j{y,i?,] +g2{2,i>i},
%i. «?2).{a^i. Pi) +Pi{y>Pi] + 9i{2,pi} + {a'2.i>2} +Pi{yy P25 + 92(2. qi}> 0,
(5)
Here {p^, q^}^ for instance, is written for 3(/i,^)/3(Pi,g'i), and eveiy fifth row is
numbered.
§ 16. In order, then, to solve the equations /^ = 0,^ = we have to form such a
linear combination of the determinants of this array as will be a complete bidiffer
ential, say <l{f^, f^, f^, f^ being such functions that the equations /^ = :=^fc^^f^ = aj,
f^ = a^ can be solved for jOj, q^, p^, q^. The array contains twentyeight rows, but
thirteen of these are combinations of the other fifteen. For instance, multiply the first
row by dfjdx^, the second by dfjdj/y the third by S/J/S^, the seventh by 3/i/3pi, the
eleventh by dfjdp^y the fifteenth by Sfjdqiy the nineteenth by 9/i/9g2 a^d add ; the
resulting row is
^(^i/i), 0, 0,
which vanishes. Other vanishing rows may be formed similarly by combining the
rows of the array so as to have in the first column one of the following —
(^i^iJll d{x.^J\)> d{yj^\ d{z,f^\ d{p^Jl)y d{p^yfl)y d{q^Jl)y d{q^yf\)y
%i>/2)> d{^^2yA\ %>/2)> d{zj^\ dip^yAl d{p^J^\ d{q^J^\ d{q^J^).
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PARTIAL DIFFERENTIAL EQUATIONS. 165
The coefficients in these combinations are partial derivatives of f^ or ^, thus, for
instance,
d^vM = ^ rf(Pi. ^.) + ^ d{p,, X,) + I rf{^i, y) + I rf(;>„ z)
and so in other cases.
The number of these combinations is sixteen, but it is to be lowered by three,
since d(f^^f^ and d{fci,f^ are identically zero and (^{fi,/^) can be formed by com
bining the sixteen in two ways, so that three linear combinations of the sixteen
bidiflferentials vanish identically.
Hence the array contains virtually only fifteen rows (28 — 13) and as there are
three columns, we have thirteen bidifferential expressions to combine. Any pair of
the four Amctions ajj, ^o, y, z will satisfy the two auxiliary equations, as is clear either
from the equations themselves or from an examination of the matrix ; of course these
solutions of the auxiliary equations will not give a complete primitive.
§ 17. If a complete primitive has been found it leads, as has been explained, to
four equations
Ui = «!, Wjj = a^y ^^3 = ag, u^ = a^,
and any pair of these must satisfy the auxiliary equations. Thus twelve pairs of
functions satisfying these are known, namely
a:,anda:^ {ij= 1, 2, 3, 4)
u, and Uj {i,j = 1, 2, 3, 4).
'These, however, are not all independent, but one pair is a bifunction of the other
eleven.
For if <^(a?i, x^, x^, x^, a^, a^, ag, a^) =
<^(a?i, X.2, x^, x^, tti, flo, ag, aj = 01
^{x^, 0^2, ajg, x^y a^, ag, ag, a^) = oJ
(6)
are the equations of the complete primitive, they must reduce to identities when
t/i, U.2, u^y u^ are substituted for a^, ao, ag, a^ respectively.
Hence
4>{^ly ^Z. ^3» ^J» '^l' ^2> '^3> ^^i) — ^ \ /yx
^{x^, aa, ajg, x^y Uj, W2, u^y W4) = j
Identically, and ^ 34> ^ = . ^ ^* du,,
t^dxi = — 2 >.* diHy
i oxi i aui
and the bidifferentials of the twelve pairs of functions are connected by a linear
relation.
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166 MR A. 0. DIXON ON SIMULTANEOUS
§ 18. The method of Charpit for a single partial differential equation of the first
order shows how all solutions may be deduced from one complete primitive, and it is
a question of interest and importance whether there is any analogous method for
simultaneous equations. Now it follows at once from the conditions for a complete
bidifferential that a bifunction of the pairs that can be formed from m functions, say
u^, Wg . . . . Wm, will be a pair of functions of u^ . . . . Um In the present case a
bifunction of the six pairs that can be formed with u^y t/^, t/g, u^ will be a pair of functions
of these four, and the complete primitive to which it will lead will be the same as
that given by t^i, w^. For when a solution of the auxiliary equations is known it leads
directly to one and only one complete primitive by the integration of the equations*
^y = Pi ^^1 + Pz ^^2
dz = q^dx^ + 9^2 ^^2;
also the complete primitive to which the equations F^ {u^y t/g, u^y u^ = const.,
F2 (ti], t/2, ^^3, u^ = const., will lead can be no other than is given by
u^ = ai, 1^2 = ^2» ^'3 = ^s» "^4 = ^4
It must not, however, be forgotten that the system F^ = const., F2 = const.,
y^ = 0, /2 = may have a singular solution. If F^, F2 involve two other arbitrary
constants this singular solution will involve four, and therefore in general be a com
plete primitive of the equations yj = 0, ^ = 0. Moreover, all new complete primitives
are included among those thus given.
For every solution implies six equations connecting x^^ 0^2, y, z^ Piy qi, p^y q% (two
of these six are of course f^ =: 0, f^ = 0), and, therefore, by elimination of
^1, ^2y y^ ^> Pi> 9^1 > i^2> ?2> ^^^ equations or more connecting Mj, Uc^^ %, w^, which are
known in terms of these eight quantities. If Wj, u^^ u^y w^^are connected by four equa
tions they are constants, and the solution is therefore included in the old complete
primitive. Let us, then, suppose that w^, Wg, u^y ^4 are connected by two or by three
equations,
F>i, W2, 7^3, u^) = (a = 1, 2 or 1, 2, 3).
Now if pi, pc^y g^i, g^2> are all expressed in terms of jo, q, two of their number, and
^i> ^2j 3/> ^y ^y ineans of the equations /^ = 0,y^ = 0, the expressions
^y "" Vidxi — pc^dxc^y dz — q^dx^ — qc^dxc^
must both be expressible in the form
K^dui + A^du:^ + Agdwg + A^du^y
* Otherwise thus — ^if in the auxiliary equations we suppose /s to have the known value Ui, they become
a pair of linear equations for /4, which must be satisfied by u^, Us, U4 ; now two linear equations in six
independent variables can only have four functionally independent solutions, and one of these is known,
namely, Wi. (In exceptional cases the two linear equations for Wg, ih, u^ may be equivalent ; for instance,
suppose fi = pi + qiy Ui = p2 + <tiyh having any form.) Hence, except in special cases, the particular
complete primitive is defined when one of the "functions %\y t*2, t^, w^, or more generally a combination of
them, F (Wj, w^? ws, W4) is known. In the case supposed in the text two such combinations are known.
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PARTIAL DIFFERENTIAL EQUATIONS. 167
and since dp^ dq are absent we must have in each case
2 A,v: =0, t A,z = 0.
r=I op r=l Oq
Thus.the equations
dy = p^dx^ + pcidxo^ dz = q^dx^ + g^^^aj^
become
1 dii^y du^i du^^ du^f
dui Bi/g du^ du^
^' a^' 3^' 3^ = 0.
3^1 3m2 9^8 du^
a^' 3^' a^' 3^
These two equations, connecting dui^ dii^^ du^^ du^y taken with the system
* 3F
2 ^ dur = (a = 1, 2 or 1, 2, 3),
show that if Wj, w^, 7^3, U4 satisfy by themselves no other relations than F« =
(a = 1, 2 or 1, 2, 3) we must have, as a consequence of the equations of the
solution,
r=l dlCr 3p ' r=l 3Wr S?
If, then, there are two equations
Fi = 0, F2 = 0,
the four equations
S^'=0.S 1^1^=0 (.= 1.2)
must reduce to two only. This will be the ordinary case, and we see that if the
forms of Fi, Fg, have been found by any means, the solution is completed without
integration ; the process corresponds to Charpit's method of deducing all complete
primitives from one, but it diflfers in that the functions F^, F^, are not arbitrary ;
they must, in fact, be so chosen that the four equations last written shall reduce to
two, and the conditions for this are clearly very complicated in general, though in
particular cases available forms for Fj, F^ may be seen on inspection.
In the more uncommon case, when there are three equations
F, = 0,F„=0,F3=0,
the six equations
ii 3;^ V "" ' ii 3z.. "3^  (a  1, 2, 3),
must reduce to one only.
These two cases are further discussed, from a somewhat diflferent point of view,
in §§ 21—23.
It should not be forgotten that the form in which tlie new complete primitive has
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168 MR. A. C. DIXON ON SIMULTANEOUS
just appeared is not that in which complete primitives were discussed in § 14, since
the equations are not here supposed to be solved for the arbitrary constants.
§ 19. In addition to the six pairs {u,, Uj) of functions satisfying the auxiliary
equations, we have also the six pairs (oc,, Xj) ; of these twelve, eleve i are indepen
dent, the other being a bifunction of them. If we can find : bifunction of the
eleven pairs which is not a bifunction of either set of six it Avill give a new complete
primitive ; whether every, or indeed any, other primitive is thus given is a matter
for further inquiry.
Suppose Vi = hi {i = 1, 2, 3, 4) to be a new complete primitive, then it gives six
more pairs of functions satisfying the auxiliary equations, and thus we have in all
eighteen pairs. The bidifferentials of these must be connected by (18 — 13) five linear
relations, one of which has been written (8) ; by means of the other four, an
expression of either of the following forms —
Ad(v2, t's) + Bd{v^, Vi) + Crf(t'i, r.),
can be found which will be equal to a linear combination of the twelve bidifferentials
d {xiy Xj) and d {uiy uJ). It is natural to ask whether, conversely, any linear combina
tion of these twelve which can be written in one of the above forms will lead to a
complete primitive ? In the first case this is not so, for if we take any fiinction
whatever, tj, of six independent variables, fj, ^^ . . . fg, we may choose the coefficients
<*!>••• «6> so that
shall be a linear combination of eleven* given bidifferentials ; the expression t, oLi d^i
may then be reduced to three terms, P^ dt,^ + jSg dt^ + ^Sg djg, so that for an
arbitrary function {t}) a combination of the eleven given bidifferentials can be found
of the form fi^ d{r)y £]) + ^82 d{yi, t,^ + ^Sg d[yiXz)> which is the same as Ao?(i;i, v^ +
Brf(t;, Vg) + Cd(vi, V4). This argument does not apply to the second form
Ad(t?2, ^^s) + M^3. '^i) + C!c/(vi, v^\
and further investigation may show that any combination of the eleven that can be
reduced to this formf will lead to a primitive.
* Not of any lower number in general, since the most general bidiiferential expression in this number
of variables contains fifteen terms, while the expression just written vanishes identically if
so that there are virtually only five coefficients, of which one must be left arbitrary.
t The conditions necessary that a bidifferential expression may be reducible to this foim include
algebraic ones which are the same as for a complete bidifferential, since
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PARTIAL DIFFERENTIAL EQUATIONS. 169
§ 20. Before we can claim in any sense to have found the general solution of the
auxiliary equations, we must be in possession of thirteen pairs of functions satisfying
them ; we have only eleven when we know one complete primitive, and hence one
more complete primitive, or even possibly two, must be found. An example (below,
§ 29) will show that one more is not always enough.
It is perhaps worth while to remark that any complete primitive defines the
whole system of solutions, since it defines the differential equations.
§ 21. The question of finding new solutions when a complete primitive is known
may be attacked by the method of varying the parameters. Take the equations
(6) or (7) of § 17. The problem is then to find such variable values for Wj, u^y Wg, u^
as will satisfy the equations
S P^dui = 0, 2 ^^dui = (9).
Since all variables are supposed functions of aj^, cc^, we may make one of two
suppositions with respect to w^, t^g, u^, u^ ; either they are connected by three
relations and are all functions of the same variable, say ty which is ot course a
function of ar^, Xg, or they are only connected by two relations, so that two of them
may be taken as functions of the other two.
Suppose first that they are all functions of the one variable t Then, generally,
the four equations (7), (9) will define a;^, Xg, ocg, x^ also as functions of ^, and hence
this supposition is not admissible unless it is possible to choose the functions of t in
such a way* that the four equations (7), (9) will be only equivalent to three. The
If these conditions are satisfied by an expression
it can be put in the form
2 Ai//(a'i, Xj),
t,; = 1.2. . 6
and then it must further be possible to express
6 8
2 ki dxi and 2 /i^ dxi
i=l i=l
as linear combinations of three differentials, dvi^ dv^y dv^ The discussion of the conditions therefore
belongs to the theory of the reduction of two such expressions, that is, of the extended Pfaff problem.
* It seems obvious that this will not generally be possible ; but it may be well to give an example.
Suppose the complete primitive to be
y = axi^ + bx2 + c, ^
z = cxi + ex2^ + bxiX2^j^
so that the differential equations are
y = ipi^i + p^ + qi  pix^,
z = qiXi + ij2a"2 ~ P^iX2\
then the variations of the pammeters a, &, c, e must satisfy the equations,
Xi^da + xodb + dc =
Xidc + x^de + XiX^dh = 0;
VOL. CXCV. — ^A. Z
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170 ME. A. 0. DIXON ON SIMULTANEOUS
number of conditions, which will be of the nature of ordinary diflferential equations,
thus imposed on the four parameters must not be greater than three ; for if they
are subjected to four conditions they are made invariable ; it may be, however,
less than three. For instance, a complete primitive of the equations jpi=:p2» 9i = ?2
is given by
y = a{x^ + Xg) + 6, 2; = c{x^ + 3^2) + e ;
the equations given by varying the parameters are
{x^ + Xc^)da + c?6 = 0,
(xj + X2)dc + de = Oy
which give the single differential equation connecting the parameters
da de = db dc.
We may then assume arbitrary forms for two parameters in terms of a third, and
find the fourth by integration. Say, for instance,
b = <f>{a)y c = V^(a),
then e = J<^'(tt)i//(a)da,
a^i + a:^ = — it>\a) ;
thus we arrive at the known general solution
y = X(^i + ^2)> ^ = ^(^1 + ^2)
whence, by elimination of xi,
{x^Hh + dcy{x^b + dc) + x^iHe^da = 0.
This equation must fail to define 0^2, so that 5, c, and a ov e must be constant ; thence it follows that all
four parameters must be constant.
I lay stress on this, because it is not in agreement with the results of Professor Konigsberger (' Grelle,'
vol. 109, p. 318), and appears in fact to show that his method there given is faulty. Professor Konigs
berger assumes (p. 313; I take m = 2) that the most general integral of the equations
f\{^\y a^2, y, z, pi, JP2, qu ^2) =
Mxu X2, y, z. Ply p2y qi, q^) =
has the form
y = wi(ari, iC2, </>i[^i(a^i, iCa)], <h[h{^u ^2)])
z = ft>2(ici, X2, </>i[^i(a^i, 0:2)], </>2[^2(a:i, a^i)]),
where </>i, <^ denote arbitrary and ^1, ^2 definite functions. But suppose these equations solved for
<^, <^2 iu the form
^\\^\{^\y «2)] = Xi(^i» ^h y, ^)
H^i(xiy »2)] = X2(«i, ^y y, ^)
and the arbitrary functions eliminated by differentiation. The differential equations thus formed are of
the first degree in ?i, ^2, S'l, ?2» *^^ *^® ^^^ V ^"7 n^eans of the general form assumed. The differential
equations in the examples given by Professor Konigsberger are, in fact, linear (see pp. 319, 328). The
method appears to be founded on an interpretation of the last clause of § 2 (p. 290), which is not justified.
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PARTIAL DIFFERENTIAL EQUATIONS. I7I
In the case of two equations of Clairaut's form
y = Pi^i + Ih^^ + <^(Pi, i>2. 9^1. 92).
z = q^x^ + q^x^ + y^fip^, p^, q^, q^\
which will be more fully considered later, the number of differential relations among
the parameters is twOy so that one parameter may be taken as an arbitrary function
of a second, and the other two found in terms of the second by solving two ordinary
differential equations.
If the primitive* is
y = aa + 6)8 + cy + ^8
^ = Aa + B^ + Cy + E8,
where a, 6, c, e are the parameters. A, B, C, E known functions of a, 6, c, c, and
^y Py y> 8 known functions of x^, Xc^, then the variations of the parameters must
satisfy the relations
cda + fidb + ydc + Sde = 0,
cudA + fidB + ydC + SrfE = 0,
and thus, in general, if a, &, c, e are all functions of one variable they are connected
by three relations
dA/da = dB/db = dC/dc = cflE/de.
The integral equivalent of these equations consists of three relations connecting
a, 6, c, e with three arbitrary constants, and by eliminating a, 6, c, e we find a new
solution of the original differential equations which is not a complete primitive,
since it only contains three arbitrary constants.
These examples show that the number of conditions to be fulfilled by the para
meters when all four are taken to be functions of one of them, may be one, two, or
three ; this number is to be made up to three by assuming arbitrary relations (two,
one, or none, as the case may be).
§ 22. Usually the parameters will not be functions of one variable only, and we
may suppose two of them, Wg, u^, to be functions of the other two, w^, u^.
The partial differential coefficients
du^ du^ du^ du^
dill * du^ ' du^ ' dtt^
are then given by the equations (9), each of which is equivalent to two. The first,
for instance, gives
3<^ 90 duj^ dcf) du^
dui dic^du^ dti^^dtii '
3^ 9^ du^ d<f> du^
9^2 du^du^ dic^du^
The derivatives are thus given in terms of le^, w^, Wg, u^, iCj, x^y ajg, x^, and the last
* It is imnecessary to give the differential equations.
z 2
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172
MR A. C. DIXON ON SIMULTANEOUS
four may be eliminated by means of the relations (7) ; so that in the end we shall
have two relations connecting w^, u^^ Uo, u^, and the derivatives ; the problem is of
the same form as the original one, to solve two simultaneous partial differential
equations in two dependent and two independent variables.
Interchange of Variables and Parameters.
§ 23. A curious thing may be noticed at this point. If in the equations <^ = 0,
^ = 0, we treat x^, x^^ Xz^ x^ as arbitrary constants and eliminate them by differentia
tion, we are led to the same differential equations connecting w^, Wg, W3, u^ as were
just now given by the variation of parameters. Thus two equations in two sets of
four quantities will give two pairs of simultaneous partial differential equations by
taking each set of the quantities in turn as variables and the other as arbitrary con
stants. The auxiliary equations, if expressed in terms of the eight quantities, will
be the same in both cases ; this gives a meaning to the six solutions of the form
(x,, Xj) which we found the auxiliary equations to have, for any one of the six
wiU lead to the primitive <^ = 0, ^ = of the second pair of differential equations,
just as a solution {ui, uj) leads to this primitive for the first pair ; any new solution
of the auxiliary equations will in general lead to a new complete primitive for either
pair, but an exception to this rule will arise when, for instance, the x differential
equations have a complete primitive which gives three relations among u^, u^^ W3, m^.
The array (5), transformed so that the variables are aj^, x^, ajg, x^, u^y Uc^, M3, u^,
connected by the equations ^ = 0, t/r = 0, will have six rows of the form
d{xi, xj), 0, 0,
six of the form ^(w/, w^), 0, 0,
and in the other sixteen there will be
d{xi, uj) in the first column,
in the second the minor of ^ ^ in the determinant :
^<f>
d^<f>
S?<f>
3^^
30
dyjr
ctejBMi '
d^idu^ *
ari8?tj '
dxjdu^
arv
a.i
d'4>
d';f>
d^<f>
d'4,
d<t>
3^
dx^dic^ '
dXc^du^ '
dx^ii^ '
djc^ii '
Br,'
3iCjj
a««^
av
^<f>
3'*^
d<f>
Byfr
ar>i'
a^3.^'
a«s3?*3 '
a^^jSw,'
3.3'
dx^
d^4>
d'if.
3*0
5^
d<f>
d^lt
dx^Ui '
dx^d^Uj '
dx^duj'
dx^du^ '
dx,'
^x^
d<f>
dtf,
9^2 '
Btf,
9«s'
3<^
3«/
0,
3«3'
Bylr
0,
(10)
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PARTIAL DIFFERENTIAL EQUATIONS. I73
in the third the same expression with <f>, \p interchanged. The array is thus practi
cally unchanged by interchanging the sets x and w, as should be the case.
§ 24. This transformation may be accomplished by taking the equations
from which may be deduced
d_ I d^ du{\ __ ^ /^ 9<^ dUi
dx^ \ i dui dxj dx^ \ i bui d(x^
^^duiduj ^ff> duj d^<f} duj . d^ dui
i j duidi(j dx^ dx^ i duidxi dvc^ i duidy ^^ dx^ i duidz ^^ dx^
— ^^ ^^ ^M»rf^^j , ^ 9^<^ dUj d^4> duj 9^^ rfwi
i j dui^Uj dx^ dx^ i duid.Vcj^ dx^ i duidy ^^ dx^ i dicidz ^^ dv^ '
Now Pi, q^y p^y q.^ are given by the relations
and hence this equation may be wiitten
* \dx^ d{x^,y,z) J i \dx^ d{x^,y,z) J
(11);
in this if), xff may be interchanged so as to give another equation.
Now, suppose ^ = aj, X = % to be two of the four equations connecting
^1, ^^2j ^3? ^4 with ccj, x^y which yield a new complete primitive, and that y, z have
been eliminated from ^, x by means of the equations <^ = 0, ^ = 0, then the deriva
*^^^® dx* dx' ^^'^ ^^ given by the following relations : —
^ 9<^ dUi _^ Q
i diti dxi '
^d±dui _ ^
i 9m; dxj
.d0 dUi , 00
i oiii dx^ co\
i dui dj\ D./?! '
and similarly for the derivatives with respect to x^.
Substituting the values hence found for these derivatives in the equation (11), we
have an equation linear in the Jacobians of the form
l^''y(i=l,2;J=l,2.S,4).
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174 MK. A. C. DIXON ON SIMULTANEOUS
the coefficient of the Jacobian written being the minor of g^rg^ ^^ ^^^ determinant
(10). Hence the constituents in the second column of the transformed array are as
stated, and those of the third are found in like manner. It is not, of course, neces
sary that these columns should be the same as would be found by actual substitution
of the values of pi, jo^, Ji, q^ in the columns of the original array ; a linear trans
formation is allowable, with constant or variable coefficients.
The above process gives fifteen independent rows of the array ; the others are
deduced from the consideration that y, z are known in terms of x^^ x^^ t/j, tfcg, u^^ u^
from the equations ^ = 0, ^ = 0.
Examples.
§ 25. I. As a first example of the method of solution, take the equations
ttj = ttg, ^1 = ^3,
where aj, ^S^ denote known functions of ajj, jp^, q^ and ag, fi^ known functions of
a^2> P2> ?2
In the array (5) multiply the seventh row by ^~^^y the fifteenth by g;^^' ^ L the
twentyfourth by  ^^^ ^l , and add. The result in the first column is d{aL^y fii), in the
second, by virtue of the particular forms of/^ and/gj
{^i.i^i} {?!> ?2} + K. 9i} {Q2rPi] + {Pi> ?i} {a^i» ^2} ^r 0,
and in the third,
i^i^Pi] {^2. 9^1} + {^1. qi) {PiyP2]  [Pu qi] {^1,^2} or 0.
Hence a^, ^Sj are two functions satisfying the auxiliary equations, and a solution is
given by finding p,, q,, p^, q^ from the equations
^1 = «2 = ^>
^1 = ^2 = &>
and integrating. Two constants will be introduced by integration, so that the result
is a complete primitive.
§ 26. II. Take, secondly, the equations
y = Pi^i + F (0:2, i>i, g^i, i>o, q^\
z = q^x^ + G {x^, ^1, 5i, 7)2, q^y
Here the twentyfourth row is
^bi> ?])> 0, 0,
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PARTIAL DIFFERENTIAL EQUATIONS. 175
so that pi, 9i are two functions satisfying the auxiliary equations, and the integral is
to be foimd by putting p^ = a, 5, = h. Thus we have
y — axi = ¥ (Xz, a, h, p^, q^),
z~bXi = G (x^, a, h, p^, q^),
or 7i = F{la,h,v',i%
C = G{i,a,b,y,',C),
where ^ = cCg, 7^ = y — ax^^ ^= z — hx^^
r,' = d7,/d^, i' = dildl
These are ordinary differential equations, the solution of which will involve two
new arbitrary constants and so constitute a complete primitive of the original
equations.
§ 27. HI. The equations
y = l^i^i + P^^ + <^( Ply P2> ?i. <1^\
z = q^x^ + q^x^ + V^(^i, Pa, q^, q^\
are of special interest, because more complete primitives than one can be found. The
obvious solution is p^ = a^, p^ = ag, qi = 61, q^ = &2>
y = a^Xj + a^Xc^ + <^(ai, ag, &i, 62)^
z = b^x^ + h^x^ + y^{a^, %, 61, h^.
Suppose a^ a^^ h^y h^ to be variable, but functions of one variable only— say aj, then
their variations must satisfy the relations
x^ da^ + x^ da^ + ^<^ = 0,
Xy dh^ + x^ d\ + ^'A = 0.
These define x^, x^, and, therefore, also y, 2 as functions of a^, unless the determi
nants of the matrix
da^y da^y d(f> 11
db^y db^y d^ I
vanish ; it is necessary, then, that these determinants should vanish. Thus
^u &i» ^2» ^2 ^re connected by two ordinary differential equations. We may assume
any third relation connecting them at will ; suppose b^ = ^{^1)9 ^ denoting an
arbitrary function. Then by integration we may suppose a^, &2 fo^^d in terms of a^.
Also a^ is connected with x^y x^ by the relation
^^ + ^2 da, + da, "" "*
SO that ttj, 61, a2, 62 ^^e all known in terms of Xj, oTo, and by substitution the values
of v, z are found.
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so that
176 MR. A. C. DIXON ON SIMULTANEOUS
§ 28. The solution may be verified. We have taken h^ = F(ai), a known but
arbitrary function of a^, and a^, h^ other functions of aj, such that
Then we have the further relations
which are of course not distinct. Also
P^^^^ de, [^ ^^ ^2 rfoj ^^ 3a, ^^ 3^3^01 ^ 36i da^ ^ 36, (iaj " ^^'
and in like manner p^ = a^.
Again 2? = 6i iTi + &2 ^2 + ^{<^iy %» ^i» ^2)^ and
^ « 7, I ^ L ^*l I ^ rf^g 3^^ 32fr da2 3i/^
9^1  oi i ^^ i^a^i ^^^ "<■ ^2 ^^ h 3^^ i 9^^ ^^^ "^ 36^ rfo, "^ (^6. daj "" ""^^
and similarly g^ = 6^.
Hence the original differential equations are actually satisfied. If the arbitrary
relation assumed — which may if convenient involve more than two of the parameters —
contains two arbitrary constants, the new solution will generally be a complete
primitive, since two more constants are introduced by integration. *t
* The ordinary equations to be integrated may have a singular solution with one arbitrary constant,
or with none : if the arbitrary function has been chosen so as to involve three or four arbitrary constants,
the whole number being thus raised to four, the solutipn so given may quite well be a complete primitive,
and, in general, will be so.
t The above investigation in a modified form shows how to find integrals of a system of three
equations
/2 (w, J, i?i, ;?2, gi, ?2) = 0, L (12)
/s (w, t;, i>i, i>2, gi, ^2) = o,J
where u = piXi + p^x^  y,v = qiXi + q^x^  z.
One solution is to take t^)?M'i,^2)^ii $2 £U3 constants connected by the three relations (12) ; if they are
not constants we have
du =■ Xi dpi + iCa dp^^
dv = xi dqi + X2 dq2.
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PARTIAL DIFFEllENTIAL EQUATIONS.
177
§ 29. Let us now consider the new solutions of the auxiliary equations, given by
the new complete primitive. The old solutions are the six pairs of the form a;,, xj and
the six of the form t<,, tij, where Mj = pi, % = p,, Wj = gj, tt^ = q^. The bi
differentials of these twelve satisfy the relation
d{y, z) pAx^, z) pod{x^, z)  q^d^, x^)  q^y, x^) + {p^q^  Piqi)d{x^,Xi)
+
x^
_^3^
3f
«1
3*
3j,
3^
«2
3<^
3?8
3^
3^
^'(Pi.l'a) +
<^(Pi. 92) +
«?(;>2. g'2) +
X,
.3*
3<^
3f
«1
3^
3<^
33i
3^^
3^
+ 3j.
3«
3(^
a?.'
^
Xl
3Vr
^(Pi> 9i)
^(i^2> 9l)
<^(<?l» ^2)
In the auxiliary equations we may take ajj, x^^ pj, p^^ q^y q^ as independent variables,
since y, « are given explicitly in terms of these six.
From (12) follow three more relations connecting the six differentials du, dv, dpi, dp2, dqi, dq2, so that
their ratios are determinate, and therefore w, r,^i, qi,p2, Ja can only be functions of one variable. The two
equations last written will then, generally, give xi, x^ in terms of this variable, which may not be. Hence
we must have
du = Xdv^dpi s= \dq\^dp2 = ^^2j
and since df\ = 0, c(^2 = 0, dfz = 0, and du^ dvy dpi, dqi, dp^, dq^ do not vanish, X must satisfy the
equation :
Sib dv ' dpi dqi ' dp2 dq2
\^ + ifi X^ + ^ \^'f^+^l
du dv ' dpi dqi ' dp2 dq2
du dv ' dpi dqi ' dp2 dq2 :
If X satisfies this equation the differential relations du = \dv, dpi = A^i, dp2  Mq2 reduce to two only,
since u, v, pi, ^i, p2, ^2 are connected by the equations
/l = 0,/2 = 0,/5 0.
By integrating these two we find two more relations involving two arbitrary constants. Hence, wq ^niay
suppose V, pi, p2y qi, 92 expressed in tenjils of t/, atid fin4 a solution by eliminating u from the following : — '
, u :^ piXi +p2X^  y,
V = ftJCl + 22»2  ^,
1 =. Xi dpifdu + ^n dp2ldu,
VOL, CXCV. — ^A, 2 A
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178 . MR. A. C. DIXON ON SIMULTANEOUS
Then
rf(«i» y) = pA^i> ^2) + [^x + ^J^i^uPi) + (ajj + ^J%i» P2) + g^ <H^u qi) +
and similar expressions may be found for (/(ajgj y), c?(a;j, 2), ^(arg, 2) in terms of the
bidifferentials of the pairs of independent variables.
Let c„ ^2, C3, c^ be the constants of integration in a new complete primitive
found by the method of §§ 278. Let X be the common value of the ratios
dp^Jdqi^ dpjdq^, d^/d^. Then, after integrating the equations dpjdq^^ =
dpjdq^ = d^/d^ (= X) by help of an assumed relation connecting, say, jp^, g^, jpg, g^,
(?i, Cj we have four relations among
and we may therefore suppose pi, pg* S'l* 5^2 expressed in terms of X, Cj, c^, C3, C4,
unless X is a constant, and therefore itself a ftmdtion of Cj, Cj, C3, C4. Then
dpi ^ X d^Tj, dp2 — X dq^^ d<ff '^ Xd^jt
will be linear combinations of dcj, rfcg, cfcj, 0^4, and so will some such expression as
adpi + $d\^
where a vanishes if X is one of the constants or a function of them. Conversely,
rfcj, rfcg, dcg, dc4 will be linear combinations of
dpy — X d^i, c?p2 " ^ ^9^2> ^ — ^ ^^> a ^JPi + /8 ^X,
and the bidifferentials of Ci, e,, Cg, C4 in pairs will be linear combinations of the six
following expressions : —
^ {Pi> P2) ^ >^d{qv p%)  ^{Pi^ q^) + ^^^ (?i> qz)^
d (i>i, <^)  Xd (^1, <^)  Xd ( Pi, i/r) + fc«d (/?!, 1/1),
^ (i>2> <^)  ^d (^2, <^)  \d {p^, i/r) + X^d (92, i/r),
/8d (jpi, X)  akd {q,, p,)  fiXd (q,, X),
a^ (i>2> Pi) + fid ( P2» ^)  a^^ (?2» Pi)  fi^d (^2, X),
acZ(<^, p,) + i8d(<^, X)  aXdii^, p,)  0XiJ(f, X).
These are combinations of the bidifferentials of p^, p^^ q^^ q^y in pairs, with the
expremonfl
d{<^,X)Xci{iA,X).
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Google
• PARTIAL WFPEBENTIAL EQUATIONS. 179
Now X is a definite function of ojj, aug, pi, p^, qi, q^, given by eliminating the
differentials from the equations
dpi s= \dq\> dpi ^ A<ig2, d^ = )di^.
By means of the first two, the third becomes
and the fourth
The result of ^^lioMiifiAion i« th^rofoise
This shows the form of X as a function of asj, jCj, ^Ji, jjj, ^j, ^j, not involving Cj, Ca,
Cj, C4. Now this choice of X makes it possible to choose coefficients A, B, C, E, F, G,
such that
^1 ^i + ^» dPi \ d<f> = A {dpi — Xdg,) + B {dp.^ — Xdgj) + C {d^ — Xdi»),
Xi dqi+x^dq^ + d^li^E {dpi — kdqi) f F (rfpa  Xtijj) + G (rf^  Xdt^).
Thus
Hd{Pi, X)  \d{qi, X)} +B {rf(i>2. X)  Xd{q„ X)}
+ C{d(<^, X)  Xdi^j., X)} = xid{p„ X) + a;ad(pj, X) + d{(f>, X)
= multiples of bidiffereatials of jp^, p^, qi, q^
+ ^{«i <^ bi. ari) + Xid (pi, x^) + rf (^, a;,)}
iW r * I
+ a^{»i <^ 0>;, .aJ«) + XafiiPp x^ + d {^, aa)
~ a^l*^ ^^' ^») ~ P^^ ^^^' ^»q "*" ^l*^ ^^' ^^^ ~ ^^'^ ^^^' ^*q
+ miiltiples of bidifferentials of ^], p^, q^, q^.
2 A 2
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180 MR. A. C. DIXON ON SIMULTANEOUS
In like manner
E {d(i)„ X)  Xd{q„ X)} + F {d{p,, X)  \d{q,, X)}
+ G {d{<f>^\)  \d (ir, X)} = ^ {d {z, x^)  g.,d (a;^, x,)} + ^{d{z,x^)  q,d {x,, x^)}
+ multiples of bidifferentials of p^, p^, g^, ^g.
Hence the three expressions
d{p,,\)^\d{q,,\),
are all reduced to the same, save for a factor, by adding or subtracting multiples of
the bidifferentials of Xj, a?^, ojj, x^ and of u^y %, 1^3, t^^ ; the same is therefore true of
the bidifferentials of Cj, Cg, C3, c^. Hence all the new complete primitives found by
the method of §§ 278 only add one to the eleven known " bifimctionally " indepen
dent pairs of functions satisfying the auxiliary equations ; one more pair, leading to
a fresh complete primitive, is yet to be found.
§ 30. These results may be used to construct examples of bifunctions. For
instance, the equations
% = q^x^ + q^^ + p^y
lead to the following case among others : —
In the equations
^ = ^ = ^^^ = X
dq^ dq^ dp^
put (/o = X + a, dq^ = rfX, and integrate.
Thus 2^2 = X* + 6, 351 = X' + c, 4pi = X* + e,
and the arbitrary constants a, 6, c, e in the new solution are respectively equal to
q,2 — X, 2^2 — X^, Sq^ — X^, 4pi — X*, where X^ajj + Xg + X = 0.
Now from § 29 it follows that d (c, e) can be expressed in terms of d (a, 6), the
bidifferentials of x^, x^, y, z and those of p^, p^y ^u 9^
For convenience, let us write
w, V, w, Xy y, z for Xj, X, py p^y qi, q^ respectively ; then
for x^ we must put — v(l + wv),
for y „ „ tw — xv{l + uv) + y,
for z „ „ yu — zv(\ + wv) + x,
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PARTIAL DIFFERENTIAL EQUATIONS. 181
so that the eight original variables connected by two equations are now expressed in
terms of six.
Thus d (3y — t;^, 4ti; — v*) can be expressed in terms of d (2; — y, 2a? — v^), the six
bifferentials of Wy a:, y, z and those of
u, — t;(l + uv\ wu — xv{l + uv) + y, yu ^ zv{l + uv) + x,
that is, of
u, V, wu — xv{l + t^v) + y, yte — zv{\ + wv) + x.
There is no difficulty in finding the relation. It is
M»(i(3y  t;», 4i^  t;4)  62;2(1 f uvfd{z  i;, 2x  v^)
 12w2rf(y, w) + 12v2(l + wi;)»d(z, x)
+ 12i;*{(l + ^v) {y — 2;t;*) ^ ^(^ — ^)}d{Vy u)
— I2i;*{l + uv)d{Vy yU'^zv '^ uz't? + x)
+ 12i^t;*c?(v, t(n^ — ajv — t^on;^ f y) = 0.
Here then we have an identical linear relation connecting the bidifferentials of
seven pairs of fiinctions of six variables. Any one of the seven pairs is accordingly
by definition a bifimction of the other six.
Second Application.
§ 31. Take now a differential equation of the second order,
where p, q are the first and r, 5, t the second partial derivatives of z with respect
to Xy y.
A complete primitive will consist of a single equation in aj, y, z involving five
arbitrary constants, say aj, ag, ^3, a^ a^. If we form the first and second derivatives
of this equation we shall have, in all, six equations from which a^, aj, aj, a^ a^ can
be foimd in terms of a:, y, 2;, jp, q^ ?% 5, ty and the original differential equation will be
the result of eliminating a^, a^, ag, a^y a^. Let w^, w^, Ug, w^, Wg represent the
expressions found for a^y a^y %, a^, % respectively, in terms of Xy y, «, p, y, r, 5, t.
Then from the equations
/=0, Wi=ai, Wgsa^,
by differentiating, we can form six equations which will involve the third derivatives
of 2 ; by eliminating these we deduce the following two differential equations to be
satisfied by Wj, u^ : —
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182 MR. A. C. DIXON ON SIMULTANEOUS
J{x, r, t) + pJ{z, r, ^ + rJ(jp, r, «) + s3{q, r, ^ + J(y, «,<) + g^^ «, e)
J(a?, «, r) + pJ(2, 5, r) + rJ{p, «, r) + 8J{q, Sy r) + %, i, r) + 5J(2:, ^, r) + sj{p, i{, r)
+tJ{q,t,r) = 0.
Here J ( ) denotes the Jacobian of /, u^, Uc^ with respect to the variables specified.
These equations express the conditions which are necessary and sufficient in order
that
dz = pdx f qdyy
dp = rdx + sdy,
dq 9 3ch + idy
may be integrable without restriction, when r, 5, ^ are given in terms of Xy y, 2;, ?, g',
by the equations
the conditions must of course be satisfied by any three of the six functions Wj, Wg,
tig, %, tig, yi We thus have forty equations, of which only eight can be algebraically
independent
I '32. The conditions to be satisfied by ti^, u^ are linear and homogeneous in their
Jacobians with respect to the eight variables x, y, 2J,p, g, r^ s, t; of these, one is
given in terms of the rest by the equation /= 0, and may, if convenient, be sup
posed not to occur in u^^ u^ : hence the auxiliary equations in this case have seven
independent variables and the dependent variables do not occur explicitly: to
find a solution we are thme&r^ to Soim a OQmpkte bidi£S^WPtial, whioh ^all bp a
linear combination of the determinants of the following array : —
d{r,s),
X^ZrP
sQ
d{r,ff,
Ti + pZ + rf + iQ,
_ Y  ^  ^P 
tQ
d{s,t),
Y + qZ + sP\ tQ,
• (^P.^,
rT,
rSAT
(5) d{p,s).
*T,
rJR
dip.,tl
 rB  «S
sB.
«?(2. r).
sT,
StT
d{q,s),
tT,
sR
d{q, t),
 sR  <S,
tR
(10) d{z,r),
i>T,
pSqT
d{z,s).
?T,
pB.
#,<9.
/)R^,
^R
d(x, r),
T.
S
d{x, s).
0,
R
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PARTIAL DIFFERENTIAL EQUATIONS. 183
(15) 4x,<),
R
^.r).
0,
T
%.*)»
T,
%.o.
s.
R
Ax,p),
0.
(20) d{9,pl
0,
<H?^>pl
0.
0
rf(x,Qr),
0,
%.^).
0,
d{z,q),
0,
(25) d(p,g),
0.
ci(x,«),
0.
%.^).
0,
d{x,y\
0,
X,P. .
. are written for
df/dx,
,^fl^ . . .
Of these twentyeight rows, only twentyone are independent* Fw instanoe,
multiply the Ist, 2nd, 4th, 7th, 10th, 13th, 16th by  S,  T, P, Q, Z, X, Y respec
tively and add ; the resulting row is
d(/r), 0, 0,
which vanishes since f=Ohy hypothesis.
Suppose d {uyy u^) to be the complete bidifferential formed from the determinants
of the array, then to complete the solution we have to find r, 5, t from the
equations
and integrate the equations
dz = pdx + qdy, dp = rdx + sdy, dq = sdx + tdy.
It will amount to the same thing if we treat t*i as known in the auxiliary
equations. They must be satisfied if Wg, u^, u^ are substituted in turn for tig. Now
two homogeneous linear partial differential equations in seven independent variables
can at most have five common solutions, and here one of these, tt^, is known ; the
other four may be taken as u^, t^, ^4, u^.
§ 33. Any two of the five ftmctions x, y, 2, p, q will satisfy the auxiliary equations,
but as we have to solve for r, 5, ^, these solutions will not serve our purpose. They
are ten in number, and ten more will be given by taking in pairs the expressions ti^,
^2> ^'3> ^4> '^5 given by any complete primitive. These twenty are not all bifimc
tiotially independent, for since there are three relations* among the ten expressions
a?, y, 2J, p, q, Wi, 1/2, W3, «4, z^6,
* Compare § 34> {k 184.
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184 MR A. C. DIXON ON SIMULTANEOUS
three linear relations can be formed connecting the twenty bidifferentials ; one is
formed from each pair of equations as at § 17 (8). Hence seventeen biftmctionally
independent solutions of the auxiliary equations are known when we have one com
plete primitive. The ftdl number is nineteen (t^ " 2 j, and in order to know all
we must have one, or possibly two (see § 41, p. 190), more complete primitives.
§ 34. New solutions found by varying the parameters may be divided into two
classes, according as the parameters are or are not all fiinctions of one variable ;
solutions of the former class only occur in exceptional cases, and the principles of § 21
apply to them with slight modification.
Let the three equations connecting
«i Vf 2J, p, q, tti, Wg, ^8. ^4* ^6
be <i>t{x, y, z, p, q, u^, u^, Wj, W4, u^) = (t = 1, 2, 3) ;
(the forms <^, <^2, ^g are not unrestricted, but must be such that the following rela
tions hold identically
or we may take <^i as not involving />, q and ^^^^ P d<f>Jdz + 9<^,/3ic
then the variations of the parameters must satisfy the three equations
[l^^d^r ^ {i = 1, 2, S),
in order that the same relations may subsist among a?, y, 2, p, q, r, s, t and the para
meters, as held when the parameters were constant.
If the parameters are functions of one variable, their forms must be so chosen that
the three equations last written reduce to one only, otherwise we shall have five
relations connecting x, y, z, p, q with this single variable.
§ 35. If the parameters are not functions of one variable, only the equations
are equivalent to six, and determine the partial derivatives of v^, u^ u^ with respect
to Wj, % in terms of the five parameters and aj, y, 2;, jp,, q. By help of the relations
^i = we may suppose x, y, 2, p, q eliminated and thus arrive at a system of four
partial differential equations connecting Wj, u^y w,, w^, u^.
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PARTIAL DIFFERENTIAL EQUATIONS. 185
The original system may also be taken to consist of four equations connecting five
variables a:, y, z, p, 9, namely :
dzjdx = p, dzldy = 5, dp/dy = dq/dx
f{^y y. 2J, p, g, dp/dx, dp/dy, dq/dy) = 0,
and so the method of variation of parameters does not lead to any simplification of
the problem in general.
§ 36. The interchange of variables and parameters is again possible ; it is, perhaps,
made clearer by taking three equations of perfectly general form,
<f>i (o^i, x^y x^y x^, x^y u^y u^ u^, u^, Ug) = (^ = 1 , 2 , 3),
connecting two sets, each of five quantities.
Whichever set we suppose constant and eliminated by diflferentiation, we are led
to a system of four partial diflTerential equations connecting the quantities of the
other set, two of the five being taken as independent variables. A new solution of
either of these systems of differential equations will in general yield a new solution of
the other.
Suppose, for instance, that we have a new solution of the u equations ; this gives
t^3, u^ Wg, say, in terms of f.^, u^. Then the six equations included in
't^^' dur 0(i= 1, 2, 3)
give two relations among cc^, . . . Wj, t^^, since the four differential equations,
which are consequences of these six, are supposed satisfied ; by the help of these
two, u^y u^y may be eliminated from the three relations <^i = 0, ^^ = 0> ^3 == ^> ^^^
thus three relations are given connecting aj^, iCg, x^y x^y x^ ; these three will constitute
a solution of the x system of differential equations.
§ 37. In this more general case there will not seemingly, as a rule, be any more
solutions for either system of differential equations. For the derivatives, say, of
^> ^4> ^5 with respect to x^y x^ are given in terihs of these five variables and two
others, say Mj, u^. The forms we may assign to u^y u^ are then restricted by three
differential equations derived fi'om the three conditions
i^ J^'^L^ /'r — 3 4 ^)
dx^ cLc^ ■" cU^ clx, ^ "" ""' ^' ""''
and thus, generally speaking, no forms of W, iig will be suitable. In some cases the
conditions are not inconsistent, and we may form an array by the method of § 11
such that if d{0yx) is a combination of its determinants, then ^ = a, x = &> ^1 = 0.
VOL. CXCV.— A. 2 B
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186
MR. A. C. DIXON ON SIMULTANEOUS
<^2 = 0, ^3 = Mrill give suitable values for Wi, Wj. This array will have four columns
and fortyfive rows, ten such as
d{xi,xj), 0,0,0,
ten such as d (m„ uJ), 0, 0, 0,
and twentyfive of the following type. ■ In the first column there is d{xi, Uj), in the
(r + l)th the minor of S^<l»r/dx,duj, in the determinant
d^<f>r a^r 3'^r
^<l>r
3*^r
d.i\ dtii 3^3 dii^' Sij 3mj' B'j 9«j' 8/j 3j<j
3'4»r
3dii
3r^
3^s
3^1
3^
3*3
3^8
3^
3«8
3^
3^3
3*3
3tB,
for
3^
3^,
3^3
3c^
r = 1, 2, 3,
3^<^,
a^j 3»<j
3^
3<ib
308
3^5
3<j^
3^
3^8
3*1
3z^3
3ii
3^4
3^
3l^6
3*2 3*3
9mj 3?^i
3*3 3^
3*2 3*3
3ws 3i^3
3*2 3(^g
3*2 3^
3^6 3i^6
This interchange of variables and parameters may take place whenever their
numbers are equal, the diflferential equations being of the first degree.
Eooamjyles.
§ 38. L As an example of the method of solution take the equation a = /8, where
a is a function of r, 5, p — sy^ x and fi a function of s, t^ q ^ sx, y.
In the array (§ 32) multiply the first row by 9a/9r, the fifth by 9a/9jt>, the fourteenth
by 9a/9a:, the seventeenth by — 5 9a/9p, and add ; the resulting row is
d{a, s), 0, 0.
Hence we take a = /8 = a, 5 = 6,
z = hxy + X + Y,
X being a function of x only and Y a function of y only. Then a = a is a relation
connecting x, dX/dx, d^X/dx^, and fi = a is a relation connecting y, clY/dy, d^Y/dy^^
and by solving these for X, Y respectively we shall have the complete primitive.
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PARTIAL DIFFERENTIAL EQUATIONS. 187
§ 39. IL As a second example take the equation
F(r, s,t,p — ^,q — ty,z — qy + \tif^, or) = 0.
Here the third row of the array is
d{s, t\ 0, 0,
so that the fiinctions «, t satisfy the auxiliary equations. Put, then, 5 = a, t = 6 ;
thus
gr = owr + 6y + c
2 = aa?y + i 5y» + cy + X,
the last term heing a function of x only. The differential equation thus becomes
F{d^X/clx\ a, 6, dX/dx, ax + c, X, x) = 0,
an ordinary equation of the second order giving X in terms of x and two more
arbitrary constants ; hence the finding of a complete primitive is reduced to the solu
tion of the equation last written,
§ 40. ni. If the equation is of the particular form F(r, s, t,p ^rx ^sy,q — sx^ ty,
2: — px — qy+^rx^+sxy+^ty^) = 0, the first three rows of the array are
d{r,s)
d{r,t)
d{s, t) 0.
Hence any two of the three functions r, s, t will satisfy the auxiliary equations,
and a complete primitive is given by putting
r ^= a^ s = h^ t = b.
Hence p = ax+hy'\g, q = ^+6y+/
z = c + gx +Jy + ^{aa^ + 2hxy f bf),
where a, 6, c,/, g^ h are constants satisfying the relation
This is a case in which other solutions are readily given by supposing the para
meters variable and functions of one variable only, say a. The variations must
satisfy the conditions
x^da + 2xydh + yHh + 2xdg + 2ydf+ 2dc = 0,
a: c?a + y rfA + dgr = 0, « dA + y d6 + cZ/ = 0,
2 B 2
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188 ME. A. C. DIXON ON STMITLTANEOUS
whence follows xdg'\ydf\2dc — 0, a simpler relation that may be taken instead of
the first of the three.
These equations will define x, y in terms of the single variable a, unless all the
first minors of
vanish.
dkt
dh
dg
dh
dh
df
dg
df
'idc
We thus have three ordinary differential equations connecting a, 6, c, f^ gyh\
they are connected also by the relation F (a, ^, 6, g, /, c) = 0, and the fifth relation
among them may be chosen arbitrarily, so that we may put h = (j) (a), an arbitrary
function.
Then we have
db/da = {f (a)}^ df/da = f (a) dg/da,
2dc/da = {dg/daf,
F{a,<f>{a),b,g,f,c) =
as the equations determining 6, g^ /, c in terms of a. These are to be integrated,
and then a is to be eliminated from the equations
03 + y dhjda + dg/da = 0,
z = c + gx +fy + ^{ax^ + 2hxy + hy^).
The result of elimination will be a solution of the differential equation. Three
constants are introduced by integration, and thus, if the function ^ involves two
constants, the new solution will generally be a complete primitive.
§ 41. The new complete primitive gives new solutions of the auxiliary equations
which we shaU now examine. Let a^, ag, Og, a^, a^ be the new set of parameters.
Then a, h, g, 6, c, / are connected with these parameters by five equations, one of
which is the original equation F = 0. These five relations are such, that if
dh = Xrfa, dg = /jida ,
then db = \^da^ 2dc = /ji^cZa, df = Xfxda ;
of these five, the first two define X, fx in terms of a, a^, a^, ag, a^, ag, and the others
must then follow from the five equations that give A, g, 6, c, / in terms of a and
the same new constants. Thus, in general, we may suppose a, A, g^ 6, c, f, /ji,
expressed in terms of X, a^, a^, ag, a^, a and the expressions will be such that
dh — \da^ dg — [xda, db — \^da, 2dc — fjL^da, df — Xfida
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PARTIAL DIFFEREXTIAL KQrATIOSa 189
involve only the differentials of «!, a^ fc^, •♦* *5 ^^ ^f these five is expites^hle in
terms of the other four, since
cF cF ?F rF rF rF
da * ck ' c* o/ ^ </^ • flr
while one of the relations connecting X, ft, a, 6, ... is
aF , ^ar , pf , ^.?f , ^ aF , , ^?f
Some expression such as vd\ — pda will also involve the differentials of a^, 04, a^,
a^, ttj only. Hence the differentials of aj, a^, o^, a^ a^ will be linear combinations ot
vdX — prfa, dh — Xc/a, rf^ — firfa, c/6 — Xt/a, rf/*— X/jw/a, 2dc — fiVa, of which the
last five satisfy a linear relation.
Thus the bidifferentials of a^, a^, a,, a^, a^ in pairs will be linear combinations of
the bidifferentials of a, 6, c, /, gr, A (only five of the six need be used) in pairs^ and
of the expressions
d{h, X)  Xrf(a, X), %, X)  ^d(a, X), c/(5, X)  X«rf(a, X),
d{f. X)  X^rf(a, X), 2d(o, X)  /^^^(a, X),
of which last five, only four are independent.
Now X, II are connected not only by the equation
dF dF dF dF dF QF
^ + ^a^+ '*^+ ^V+ ^37 + ^/^V = ^'
but also by the equation
35 + Xy 4 /t = 0,
so that they are definite functions of x, y, a, h, /, g, h.
Again p = ax + hy + g,
d{p, x) — hd{y, x) = xd{a, x) + yd{h, x) + d{g, «),
«^(p. y) — «<^(». y) = ^<^(«. y) + W(^. y) + %. y)
Thus y{<f(A, X)  Xrf(a, X)} + {%, X)  ftd{a, X)}
= xd{a, X) + yd(A, X) + %, X)
= a^ [^(p. ^)  ^«'(y. a')] + aj; I^^p^ v)  ^K^> y)l
+ multiples of bidifierentials of «, 6, c, /, g', h.
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190 MR. A. C. DIXON ON SIMULTANEOUS
In like manner
y{d{h, X)  X^a, X)} + {d{f, X)  \iid{a, X)} + x{d(h, X)  \d{a, X)j
= xd{h, X) + yd{h X) + d(y, X)
= a^ \d{<i^ ^)  &%. «^)] + ^ \A{^^ y)  ^d{x, y)\
+ multiples of bidifferentials of a, 6, c, yj gr, h.
Lastly,
2xy{d(h, X)  Xrf(a, X)} + y*{(^(6, X)  \H{a, X)} + 2ir{%, X)  ixd{a, X)}
+ 2y{d(/, X)  X,irf(a, X)} + {2ti(c, X)  /.V(a, X)}
= a?d{a, X) + 2xyd{h, X) + y^d(6, X) + 2xd{g, X) + 2t/d(/, X) + 2rf(c, X)
= 2{d{z, x) — {hx + by +f)d{y, x)}d\/dx
+ 2{d{z, y)  {ax + % + 5r)d(^, y)} d\/dy,
+ multiples of bidifferentials of a, 6, c, /, g^ h.
Hence, in all, nine combinations of the ten bidifferentials of a^, ag, ag, a^, a can
be expressed in terms of the bidifferentials of x, y, z, p^ q and of a, 6, c, fy g,h\ that
is, in terms of the bidifferentials of the seventeen known independent pairs of functions
satisfying the auxiliary equations : thus the new complete primitive adds only one to
the number of these known bifunctionally independent pairs, and one more must be
added in order to give the full number.
This theory enables us again to construct examples of bifunctions of a number of
known pairs which may reach eighteen.
§ 42. The foregoing investigation may be modified so as to give singular solutions
of a pair of differential equations of the form in question, say
Fi(r, s, t, p, q, z) = 0,
F^Cn s, t, p, q, z) = 0,
where p = p  rx ^ sy,
q = q — sx ty,
A complete primitive would be given by supposing r, 5, t, p, q, z constants con
nected by the above equations. Another solution would be given by solving the
total differential equations found by supposing the relations
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PARTIAL DIFFERENTIAL EQUATIONS. 191
xdr {y ds + dp = 0,
xds '\' y dt ^ dq = 0,
xdp+ y dq + 2dz = 0,
to reduce to the same relation linear in x and y. That is, we must solve the system
ds = \dr, dt = X^dvy dp = fidr,
where X, /x are given in terms of ^, q^ 2, r, s, t by the relations
9Fi 9Fi vaSFj 9Fi 9Fi ^ 1 o9Fi
ar + V + ^V + ^r + ^'^ar + ^'^ar =^'
9Fg 9F3 xo9F 9F3 9F3  2 9F,
and 5^, 2J in terms of jp, r, 5, i by the relations F^ = 0, F2 =0.
The complete primitive of these ordinary equations will involve three arbitrary
constants, and there may be singular solutions with a lower number ; none of these
will therefore constitute a complete primitive of the partial diiSferential system
Fi = 0, F3 = 0.
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[ 193 ]
V. The Velocity of the loris ^produced in Gases by Rmtgen Rays.
By John Zeleny, B.Sc., B.A., Assistant Professor of Physics, University of
Minnesota.
Communicated by Professor J. J. Thomson, F.R.S.
Received February 5, — Read March 1, 1900.
§ 1. Introduction.
The electrical conductivity which is imparted to gases by their exposure to Rontgen
rays has been explained by J. J. Thomson and E. Rutherford* on the hypothesis of
a formation of oppositely charged carriers throughout the volume of the gas. The
motion of these carriers or ions when in an electric field constitutes the observed
conductivity, and the recovery of the insulating property of a gas after an exposure
to the rays is due partly to the recombination of the oppositely charged ions and
partly to their impact with the boundaries.
An estimate of the sum of the velocities with which the positive and negative
ions move in air when in a unit electric field was first obtained by J. J. Thomson and
E. Rutherford, and later E. Rutherford,! by the same indirect method, determined
the sum of the velocities of the ions in a number of gases. This method involved
the determination of the rate of recombination of the ions, the saturation current
obtained through the gas by the use of a strong electric field, and the current
obtained with some small nonsaturating electric force. E. Rutherford also
describes an experiment in which the velocities of the two ions in air were obtained
separately by a direct method, and found to be approximately equal. The writerj
has since shown that in general the two velocities are not equal, and for those gases
for which the ratio of the two velocities was determined the negative ion moved the
faster in nearly all cases.
The values of the velocities of the ions have recently been applied by J. J.
Thomson§ and J. S. Townsend in the determination of important physical quantities,
and it seemed desirable that a redetermination of the values of the velocities be
* J. J. Thomson, and E. Rutherford, * Phil. Mag.,' November, 1896
t E. Rutherford, * Phil. Mag.,' November, 1897
X J. Zeleny, * Phil. Mag.,' July, 1898.
§ J. J. Thomson, *Phil. Mag.,' December, 1898.
II J. S. Townsend, *Phil. Trans.,' A, vol. 193, 1899.
VOL. cxcv— A 266. 2c 9.11.1900.
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194
MR. J. ZELENY ON THE VELOCITY OP THE IONS
undertaken, partly because of advances in our understanding of some of the intrica
cies of the conduction, and partly because it seemed desirable that a satisfactory
direct method be devised whereby the velocities of the two ions could be determined
separately, and in which the experimental conditions could be subjected to a number
of variations sufficient to ensure freedom from serious errors.
In undertaking this, an attempt was first made to use a modification of the method
employed by the writer in the determination of the ratio of the two ionic velocities,
which is described in a previous paper. The ions were made to go against a stream
of gas in a tube by means of an electric field, and their velocity was compared to that
of the gas stream. The presence of the gauzes necessary for the production of the
electric field was found, however, to disturb the gas stream sufficiently to produce a
turbulent motion in it and so prevented the attainment of absolute results.
The method which was then developed, and the one with which all of the results of
this paper were obtained, also consisted in directly comparing the ionic velocity with
that of a stream of gas, but avoided the difficulty of the above by having the electric
field at right angles to the gas stream.
§ 2. The Method Used for Determining the Velocity.
A stream of gas is passed between two concentric cylinders which are kept at
difierent potentials, and which at one place are traversed by a beam of Rontgen rays.
The ions which are produced between the two cylinders by the rays are carried
along by the stream of gas and at the same time, under the influence of the electric
force, they move at right angles to the axis of the tubes. The resultant paths of the
ions are inclined by an amount depending upon the relative value of the velocity of
the gas stream to that of the ions.
Let CC in fig. 1 represent a section of a portion of the outer cylinder, and DB
that of the inner one, and let dd represent a narrow beam of rays traversing the two
cylinders at right angles to their common axis. When the two cylinders are at
Fig. 1.
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PRODUCED IN GASES BY RONTGEN RAYS. 195
different potentials and the gas between them is at rest, an ion starting from the
point d at the inner surface of the outer cylinder will move directly across to /
imder the electric force. But when a stream of gas is passing between the cylinders
from right to left, the ion will also be carried along by the stream, and so follow a
path somewhat like that represented by the curve dk^ finally reaching the inner
cylinder at some point, i, which can be determined. The paths of the ions are not
straight lines, because the electric intensity and the velocity of the gas stream vary
from point to point between the cylinders and according to a different law for each.
The distance X that the ions have been carried along the tube by the gas stream
while they are crossing between the two cylinders under the electric force is a
measure of the relative velocities of the gas and of the ions, and so may be used in
determining the velocity with which the ions move in a given. electric field.
Let the outer cylinder be kept at a potential of A volts and the inner one at zero
potential.
Let h be the inner radius of the outer cylinder and a the outer radius of the inner
cylinder.
Then the potential at any point between the cylinders at a distance r from the
common axis of the two cylinders is
and the electric intensity at this point is
dR A
dr rlogebfa ' V /•
If we let V represent the velocity with which an ion moves when in an electric field
whose intensity is 1 volt per centim., and assume that its velocity is proportional
to the strength of the field, then at a point whose electric intensity is represented bv
equation (2), the radial velocity of the ion will be
V = ^ . (3).
The ion being carried by the moving gas also has a motion along the tubes. The
velocity of the gas stream at any point depends upon its distance from the axis of
the cylinders, which will be called the x axis.
Suppose that at the distance r from this axis the gas velocity is u.
The motion of the ion is represented by
dx u
~dT~ Y ' ' ' \ (^/'
and substituting the value of V from (3),
2 c 2
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196 MR. J. ZELENY ON THE VELOCITY OF THE IONS
The distance X travelled by the ion in the direction of the x axis while it is
traversing the whole distance between the cylinders, i.e., from r = 6 to r = a, is
^='^:'"f/^' («)•
Now the average velocity of the gas stream as measured by the quotient of the
total volume of gas emitted in a second by the area of the cross section is
From (6) and (7)
^= 2kv ^^»a • • • (^)' ^""^ ^= 2AX ^^ga " * ' ^^^^
This gives the value of the ionic velocity in a unit field in terms of quantities which
can be experimentally determined.
The time required for the ions to pass from one cylinder to the other is
The equations above apply to ions starting from the inner surface of the outer
cylinder and moving inward to the inner cylinder. In practice it is not possible to
limit the production of the ions by the Rontgen rays to the inner surface of the outer
cylinder, so a narrow beam of rays is passed at right angles through the cylinders, as
is represented by dd of fig. 1. Of the ions of this layer which move inward under
the influence of the electric force, those that start from the circumference at d are
carried the farthest by the gas stream before they reach the inner cylinder. Under
these conditions the equations obtained can be applied by determining the point along
the inner cylinder farthest from the beam of rays that is still reached by ions. For
obtaining this point, the inner cylinder DB is divided at k into two parts, insulated
from each other, the part B to the right being connected to earth, while the part D,
to the left of the division at i, is connected to a pair of the quadrants of an
electrometer.
If a definite stream of gas is maintained between the two cylinders, then while the
potential of the outer tube CC is above a certain value, all of the ions from the
volume dd which move inward will reach DB to the right of the juncture k, and so
the electrometer reading will not change. By gradually diminishing the potential of
CC a value is finally reached such that the ions starting from the outer edge d reach
DB just to the left of k, as wiU be indicated by a changing electrometer reading. The
value of the voltage A in equation (9) is thus determined, and the value of X, which
corresponds to it, is the distance from the beam of rays to the juncture k. In getting
X the corrections which must be made for the width of the beam of the rays and for
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PEODUCED IN GASES BY RONTGEN RAYS.
197
the width of the juncture k will be considered later. The apparatus as used will now
be described.
§ 3. The Apparatus.
The main parts of the apparatus are represented in fig. 2, where the lower part of
the figure is a vertical section, while the electrical connections in the upper part are
viewed firom above.
Fig. 2.
The outer cylinder, A A', had an internal diameter of 5*11 centims., and a total
length of 142 centims. For convenience the length is shortened in the figure by the
omission of two sections. The part to the left of V, 41 centims. long, and the part
to the right of V, 81 centims. long, were made of strong brass tubing. The portion
DD' between these was 20 centims. long, and consisted of an aluminium tube, which
was of the same internal diameter as the brass cylinders. Brass collars over the ends
of the aluminium tube fitted into the external collars V and V soldered to the brass
cylinders, and so formed closefitting joints that were made gastight by sealing them
on the outside. The whole cylinder was supported on a board, XX', and insulated by
means of four paraffin blocks, two of which are represented by P and P'.
The inner cylinder, BB', was an aluminium tube 1 centim. in diameter, closed at its
ends by conical pieces. At C the cylinder was divided so that the two portions were
held onehalf of a millimetre apart and insulated, by means of an ebonite plug. At
the end, B', the tube was supported and kept central by means of two small ebonite
rods, Q. The tube was further supported by the two stiff brass wires, Y and Y',
which lead through the ebonite plugs, R and R', in the outer cylinder, and served to
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198 MR. J. ZELENY ON THE VELOCITY OF THE IONS
make electrical connections. The part B' was joined to earth, while the part B w^as
connected to a pair of the quadrants of the electrometer, E. Great care was taken to
adjust the position of the central cylinder so as to be accurately concentric with the
outer one.
The ends of the outer cylinder were fitted with the large rubber stoppers F and F'.
Through these passed the gas inlet and outlet tubes, whose ends were the elongated
funnels J and J'. These funnels, together with the cone endings of the inner
cylinder, made the lines of gas motion change less abruptly on entering and leaving
the apparatus, and so aided in having the gas maintain a steady motion in DD', where
the observations were taken. At the left end, F, a rubber tube led to a gas bag of
about 150 litres capacity. The manometer, I, measured the pressure of the gas in the
apparatus. The right end, F', was connected to the glass wool chamber, G, which
served to remove dust and any stray electrification from the gas. A rubber tube then
led to a drying or moistening apparatus, to be described later, which was connected to
a large gasometer of the ordinary type. The pressure of the gas in the gasometer
was measured by means of a manometer, and a scale was also attached to the
gasometer for measuring its rate of descent during an experiment. The average
velocity of the gas stream in the apparatus was determined from the volume emitted
by the gasometer in a second, and from the area of the cross section between the two
cylinders. To prevent the gas in the gasometer from getting moist too rapidly in
those cases where dry gases were used, the surface of the water was covered with a
layer of oil, such as is used for air pumps, because of its very low vapour pressure.
The board, XX', with the attached cylinders was placed on the top of a lead
covered box, UU', so that DD', the aluminium portion of the outer tube, was above
the aluminium window, W, in the box.
The box contained the Crookes' tube and the induction coil for operating it. The
form of tube used was that which the writer has previously employed for similar
work.* This form was more satisfactory than any of the others tried, and gave the
best results when emitting weak rays, and when an interval of rest of at least three
or four minutes was allowed between the periods of use, which did not exceed thirty
seconds. A 6inch Apps' coil was used with a hammer interrupter, which could be
made to run with sufficient uniformity with an easy running weak ray tube. The
source of the rays, T, was more than 20 centims. from the axis of the cylinders.
The narrow vertical beam of rays wliich was sent up through the cylinders was
regulated by adjusting the position of the tube, T, and of the lead plate, S, with its
narrow slit, and of the two lead rings, L and U, which fitted over the cylinder, DD'.
This adjustment was first made by geometrical arrangement, and then tested and
completed with the aid of a fluorescent screen placed over the apparatus. The lead
strips, H and H', served to restrict tlie window, W, and the lead cover, Z, prevented
any rays or ionized gas from reaching the outside air of the room.
♦ J. Zeleny, a^hil. Mag./ July, 1898, p. 126.
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PRODUCED IN GASES BY RONTGEN RAYS. I99
The quadi'ant electrometer, E, used for making the measurements was a small
bicellular one, the needle of which was suspended by a quartz fibre, and charged
through the liquid below by means of a battery of 160 small storage cells. One
pair of its quadrants was joined by a wire to the part BC of the inner cylinder.
Both the electrometer and the connecting wire were surrounded by an earthed
metal case.
The key, K, permitted the insulated quadrants to be connected to earth at any
time.
The capacity of the two quadrants and the part of the inner cylinder connected to
them, together with the connecting wire, was about 53 centims. The sensibility of
the electrometer was about 500 divisions per volt, with the scale at a distance of
130 centims. The potential of the outer cylinder A A' was maintained at any desired
value by means of the battery of storage cells, N ; the arrangement of the extra
cell, O, and the divided megohm, M, permitting the addition of a fractional part of a
cell's voltage.
By opening a stopcock on the gasometer the gas was made to pass from the
gasometer, through the apparatus, into the gas bag on the other side, at a rate which
was regulated by the weights on the gasometer. It could then be forced back into
the gasometer and used again.
A large volume of gas is required for carrying out an experiment, and the method
is therefore limited to a small number of gases that can be obtained in such quantities,
and that do not act upon the materials of the apparatus.
§ 4. CORRKOTIONS AND PRECAUTIONS OBSERVED IN THE EXPERIMENTS.
1. It is essential for these experiments that in its motion down that part of the
tube where the observations are being taken, tlie different portions of the gas should
move in paths parallel to the axis of the tube, i.e., that the motion be uniform, and
not turbulent with vortices. This condition depends upon the velocity of the gas
stream.
O. Reynolds has shown* that for motion in a cylindrical tube a fluid when
started in a turbulent state wilj tend to assume a uniform motion with the parts
moving parallel to the axis when for the fluid the average velocity is less than a
critical value.
V =
A^
BpD'
where /x is the viscosity of the fluid relative to that of water at 0°, p is its density,
D is the diameter of the cylinder, and B is a constant.
The value of B obtained was about 280 when D and V were measured in metres.
Applying this constant to the gases used, for a cylinder of the diameter of the
♦ O. Reynolds, * Phil. Trans.,' A, 1883.
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200 ME. J. ZELENY ON THE VELOCITY OF THE IONS
outer one in the apparatus, we obtain for the value of the critical velocity for air.
about 55 centims. per second, and for hydrogen about 390 centims. per second. It is
evident that in the apparatus used where there are two concentric cylinders, the
maximum velocity consistent with a uniform motion must be considerably larger
than if the gas were flowing through the outer cylinder alone. Nevertheless the
largest value of the velocity used in any experiment was 25 centims. per second for
air and 44 centims. per second for hydrogen. As these values are well within the
limits given above for a cylindrical tube whose radius is equal to that of the outer
one here used, the conditions for a stable motion are fulfilled. The entrance of
the gas through a funnelshaped aperture and its subsequent passage for a con
siderable distance through a uniform section allowed the motion to come to a per
manent state before it reached the place where the observations were taken.
An experiment which was tried showed that by blowing a stream of air down a
large glass tube and with a velocity greater than that used in these experiments,
the gas assumed a motion parallel to the axis after it had traversed but a short
length of the tube, as was made visible by the presence in the air of irregularly
distributed ammonium chloride particles.
2. The volume of the gas emitted per second by the gasometer varied a little for
different elevations of the gasometer, but there was a considerable range where it
was quite constant, and this range only was used in making experiments, the rate of
descent being determined in addition during each observation. Guide wheels pre
vented the tilting of the gasometer during its descent, and the readings on the
attached scale could therefore be relied upon. The pressure of the gas was
determined by a manometer attached to the gasometer, and the pressure in the
apparatus was similarly obtained. The volume of the gas emitted by the gasometer
per second was then reduced to the pressure in the apparatus, and dividing by
the flow area in the tubes, the required value of U in equation (9) was obtained.
3. In order to understand more clearly the manner in which the values of
A and X of equation (9) were determined, let us consider the following case. In
fig. 3, CO represents a longitudinal section of the outer cylinder. DB is the
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PRODUCED IN GASES BY RONTGEN RAYS.
201
inner cylinder having the insulated juncture at ky the part D of the cylinder being
connected to an electrometer.
The gas stream is supposed to flow from right to left in the figure, and hdmn
is the beam of rays. DB being at zero potential, suppose that when the potential
of CC is at a certain value the ions going towards DB move in paths parallel to the
line ah in the upper half of the figure. An ion starting from any point to the
left of ak would reach the part D and so influence the electrometer, but as all of the
ions start from the beam of rays to the right of ak^ all of them reach B. If the
potential of CC is diminished so that the inclination of the ionic paths becomes hk,
ions from the outermost rim of hdmn will just begin to reach the part D. By a
certain decrement in the potential of CC the paths of the ions can be made
parallel to dk, so that ions will reach D from a volume whose section is represented
by the triangle hdg^ the width of the beam of rays being hd. By a decrement in
the potential of CC equal to the last one, the volume from which ions reach D is
increased by a volume whose section is seen from the figure to be nearly a parallelo
gram of about twice the area of the triangle hdg. Another equal decrement in
the potential increases the volume by almost the same amount as the last. As
the potential is diminished further, the rate of increase of volume gradually
diminishes. So if we represent the potentials used by abscissas and the volimies
from which ions reach D by corresponding ordinates, we obtain a curve, fig. 4,
R
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p/a
Cs,
/
n
/
<q
J
f
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i
i
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Fig. 4.
whose inclination to the axis of abscissas, as the potentials are increased, at first
gradually increases (RS of fig. 4), then assumes a constant value (ST) and finally
diminishes (TU) as the curve ends in the axis of abscissas. The point U corre
voL. oxer.— A. 2 D
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202 MR. J. ZELENY ON THE VELOCITY OP THE IONS
spends to the inclination of the paths of the ions represented by hh of fig. 3,
T corresponds to dk, and W to ek. As the paths change from ek to dk^ the
diminution in the number of ions reaching D is equal to about twice the number
that are getting to D in the latter case (W^ = 3 Td). If, therefore, the rate of
diminution remained unchanged until ions just ceased to reach D, the change in
potential required for this would be just a half of the change fi'om ek to dk or
from dk to hk. Thus in the curve it is seen that by prolonging WT it reaches
the axis at c, half way between 6 and d. This corresponds to a potential which
would be required for an ion starting from c (fig. 3) the middle point of the beam
of rays 6d, in order to have it just reach the juncture k in the inner cylinder.
It is evident that the points T and U are not very sharply defined on an experi
mental curve, and hence cannot be determined as accurately as the point c, and so
in practice the potential A of formula (9) has always been determined in this
latter way. Evidently the value of X which is to be used with this value of
A has to be measured from the middle of the beam of the rays where they cross
the inner cylinder to the middle of the juncture k^ as aU ions reaching the middle
point are drawn to D. The width of this juncture was only '05 centim. The
width of the beam of rays was used as small as possible, and in most cases was
•2 centim., this being a smaU part of the total distance X
4. In considering the distribution of the ions between the two cylinders while the
conduction is going on, it is seen from the lower part of fig. 3 that supposing the
external tube to be positive, the negative ions starting from 5 will describe a path
somewhat like sw, so that all of the negative ions will be confined to the space
wmnts. Similarly the positive ions starting from m will describe the path mk^
and all of the positive ions will be confined to the space kmnts. In the space
where these two overlap, i.e., omnts, both kinds of ions will be present and recom
bination will take place, the number of ions per cubic centim. diminishing, there
fore, as we go from 5m to o.
The space ovnn will be occupied by negative ions alone, and oks by positive ions alone.
wm will usually be shorter than ks^ because as a rule the negative ions travel the
faster in the same electric field.
5. Of the ions starting from m towards k all will not follow the path mi, but
some, due to the motions assigned to them by the kinetic theory of gases, will
difiuse to either side so that the distribution, along the path, of the ions which
started from m will lie between the two dotted lines mr and mp. This effect will
produce a distortion in such a curve as that shown in fig. 4, and to bring all of the
ions to the part B of the inner cylinder will require a greater force than would be
necessary if there were no diffusion. The effect of this disturbance upon the value
of the ionic velocity obtained in the manner described is to give a result that is
too small because the potential A obtained is too large. Moreover the amount of
the diffusion depends upon the time required for the ions to travel between the two
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PRODUCED IN GASES BY ItftNTGEN RAYS. 203
cylinders so that if we obtain values of the ionic velocity, in the manner already
described, these will be the larger and nearer to the true value the smaller the
time that is required for the passage of the ions across.
If this time were zero, then evidently all diffusion effects would disappear.
6. The free charges that exist in the gas, where the ions of one sign predominate,
tend to spread on account of the mutual repulsion of the charged carriers.
This produces an eflFect similar to that of the diffiision just described. It increases
with the time required for the ions to pass between the cylinders, but is less the
smaller the density of the free charges, i.e., the weaker the Rontgen rays used and
the narrower the beam of the rays.
7. The presence of these free charges in the gas also has an influence upon the
intensity of the electrostatic field between the two cylinders. To diminish this effect
a sensitive electrometer was used in making the observations, as this allowed the
employment of a weak radiation so that the charges in the gas were of a small
density.
While it is not possible to make an exact calculation of the magnitude of this effect
because of the unsymmetrical distribution of the ions, an approximation to it can still
be obtained. Knowing the capacity of the receiving system and the charge received
in a given time, and knowing the approximate velocity of the ions in the electric field
and the approximate space occupied by the firee charges, the density of these charges
can be obtained roughly, and their effect upon the electrostatic field can be
computed.
Computations of this kind made from the observations used for final results
showed that the largest value of this correction made a diminution in the electro
static field of less than 1 per cent. In some experiments where a large inner
cylinder was used the intensity of the electric field employed was less, the ions
moved slower, and the density of the fi^ee charges was therefore larger and in some
instances the above correction was perhaps nearly 2 per cent. In all cases an
increase in the strength of the field itself diminishes the percentage value of the
correction, while the simultaneous diminution in the density of the free charges
reduces it still further.
8. The motion of these free charges through the gas also produces a motion of the
gas itself, as the writer has previously shown.* The amount of this is, however, very
smaU compared to the velocity of the ions, so that it cannot have an appreciable
disturbing efiect upon the restdts of these experiments.
9. In conduction produced by Rontgen rays there is a noticeable faU of potential at
the electrodes which diminishes the electric intensity in the intermediate space. As
determined by the writer, t for conduction in air between two plates 1*2 centims.
apart, this amounted to about 2 per cent, of the total potential difference for the
* J. Zeleny, ' Proc. Camb. Phil. Soc.,' vol. 10, Pt. L, p. 13.
t J. Zbleny, ' Proc. Camb. Plul. Soc.,' vol. 10, Pt. I., p. 21.
2D2
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204 MR. J. ;ZeleNY on the velocity op the ions
strength of rays used. For the much weaker radiation and the greater distance here
used the correction does not perhaps exceed I'per cent. For gases other than air
the effect has not been determined, and has been assumed to be no greater than with
air.
10. J. Perrin'*^ has shown that when the Rontgen rays impinge upon a metal
surface the ionization in the gas near it is increased by an amount depending upon
the nature of the metal and upon the state of its surface. M. G. SAGNAct and
P. Langevin have shown since that this is due to a secondary radiation started at
the metal surface by the Rontgen rays. It is possible that the ions so produced are
of a different nature from those produced by the direct rays, but in the absence of
any evidence to that effect the much more probable case is assumed that the two
kinds are identical.
The effect of the secondary rays, therefore, is to produce an uneven distribution ot
the ions in the space exposed to the direct rays, and also to widen the ionized area
near the metal surfaces. This makes more difficult the accurate determination of the
potential A in equation (9), the tendency being to get it too large. J. Perrin found
that the surface effect was by far the least for aluminium, what he calls the coefficient
being '0 for aluminium in air as compared to '9 for gold in air. The effect is also
very much dependent upon the cleanliness of the surface. It is thus seen that in the
apparatus used this effect was made as small as possible by using unpolished
aluminium as the material for those parts of the cylinders upon which the rays
impinged. That the secondary rays did not produce an appreciable amount of
ionization at a short distance to the side of the beam of the direct rays was shown
by passing these rays near to the insulated juncture in Ihe inner cylinder while the
gas in the tubes was at rest. No conductivity was observed to that part of the inner
cylinder which was not exposed to the direct rays.
Further experiments tried for the effect of the secondary rays by coating the inside
of the aluminium cylinder on the apparatus with tinfoil will be described later among
the observations for dry air.
11. W. C. RontgenJ has shown that the air itself where it is exposed to the rays
acts as a source of a weak secondary radiation. The writer is not aware of any
experiments showing any conductivity produced by this radiation, but the experiment
referred to in the last section, where a beam of rays near the juncture of the inner
cylinder produced no appreciable conductivity on the other side, shows that in these
experiments the effect may be disregarded.
12. When D (fig. 3), the part of the inner cylinder joined to the electrometer,
takes up a charge in the progress of an observation, the electric field in the vicinity
of the juncture becomes slightly distorted, tending to lessen the number of ions
* J. Perrin, 'Comptes Eendus/ vol. 124, p. 455.
t M. G. Sagnac, 'Journal de Physique,' 1899, p. 65.
X W. C. EONTGEN, * Wied. Ann.,' vol. 64, p. 18.
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PEODUCED IN GASES BY RONTGEN RAYS. 205
reaching D. As for each reading this effect starts from zero, the only influence of
this upon a series of readings with difierent potentials is to diminish their values by
small amounts nearly proportional to their size, thus having practically no effect upon
the result obtained by projecting the curve as in fig. 4.
13. The velocity of the ions is evidently dependent upon the pressure of the gas.
In these experiments the variations in the pressure were but small, being due mainly
to the variations of the barometer. No experiments have been carried out on the
effect of pressure upon the velocity of the ions produced by Rontgen rays, but
E. Rutherford* has shown that for the conduction produced by ultraviolet light
the velocities of the ions in air are inversely as the pressure of the gas. This result
will be used in these experiments to reduce all of the values of the velocities to the
same pressure of 76 centims. of mercury.
14. The effect of temperature upon the ionic velocity is not known, so that correc
tions for temperature could not be made. The temperature was, however, taken in
all cases, so that if necessary the correction can be applied later on.
15. In considering the various corrections above, it is seen that the effect of many
of them is diminished or made negligible by using a narrow beam of weak rays, and
by using unpolished aluminium for that part of the cylinders where the rays impinge.
Those corrections which depend upon the time required for the ions to cross between
the two cylinders could be made very small by sufficiently reducing the value of this
time, but we are limited in doing so by the increase that is produced in the difficulty
of measuring one of tlie required quantities. Resort must be had to finding the
values of the ionic velocities for different times of crossing, and from these deriving
the final results.
An estimated correction of 2 per cent, will be made for those effects considered
above, especially (7) and (9), which tend to make the result too small by an undeter
mined but small amount.
§ 5. Changes made in Experimental Conditions.
The apparatus used permits of several changes in the experimental conditions,
which are a test of the accuracy of the method, and allow us to draw conclusions
about the effects of some of the corrections previously noted.
1. The velocity of the gas stream was varied by changing the weights on the
gasometer. This necessitated a proportionate change in the value of the potential A
of equation (9). The paths described by the ions are the same, but the time required
for their passage between the two cylinders is changed. There are also changes in the
amount of recombination of the ions and in the diffusion effect. The density of the
free charges is changed, and so their effect upon the electric intensity is altered, and
the spreading due to the mutual repulsion of the ions is also different.
* E. KirniERFORD, * Proc. Camb. Phil. Soc.,* vol. 9, Pt. VIII., p. 4U.
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206 . MR. J. ZELENY ON THE VELOCITY OP THE IONS
2. The distance of the beam of rays from the insulated juncture in the inner
cylinder was also changed. This likewise necessitated a change in the value of the
potential A, but in the opposite sense. The paths of the ions are now quite
different, and changes are also produced in all of the quantities mentioned in the
preceding case.
3. The intensity of the Rontgen rays was also varied. This produced alterations
in the density of the free charges in the gas, and consequently in their effect upon the
electric field between the cylinders and in the mutual repulsion of the ions. The
amount of the recombination of the ions is also affected as well as the fall of potential
at the electrodes.
4. By changing the diameter of the internal cylinder complete changes are
produced in the configuration of the forces, and of the motions of the ions. All the
other changes can also be tried in conjunction with this one.
5. The material of the inner surface of the outer cylinder was also altered to note
the influence upon the result of increased ionization at the metal surface.
6. In trying to find the effect of any of these changes upon the observed velocity
the greatest difficulty met with is due to the smallness of the effects, and their conse
quent masking by the irregularities of individual observations caused by the difficulty
of maintaining a uniform radiation for a length of time sufficient to cover a niunber
of readings. Individual observations taken under the same conditions may vary
among themselves by a number of per cent., so a smaU change in the result cannot
be detected unless a large number of observations is made.
§ 6. Method of Conducting the Experiments.
The following procedure was followed in taking readings with the apparatus. The
Crookes' tube and the lead slits were accurately adjusted, so that the beam of rays
occupied the desired position, and the distance X of equation (9) was carefully
measured. The cylinder AA' was connected to a chosen potential on the battery N.
The electrometer quadrants, joined to the part B of the inner cylinder, were then
disconnected from earth by means of the key K, and the zero reading was observed
on the scale. The reading on the gasometer scale was also taken. At a definite
time, observed on a chronometer, the valve at the gasometer was opened, so that the
gas began to flow through the apparatus. After a short period, usually 10 seconds,
sufficient to produce a steady state of flow in the apparatus, the primary of the
induction coil was closed and the rays thus started. The rays were allowed to run
for 30 seconds, and the primary of the coil was then broken, and the valve of the
gasometer was also closed at a definite time. The electrometer reading was now
taken, and the deflection produced was obtained. The key K was then closed, and
the quadrants of the electrometer were connected to earth. From the reading on
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PRODUCED IN GASES BY RONTGEN RAYS. 207
the gasometer scale the volume emitted was obtained, and with the aid of the
pressure readings which were taken the average velocity of the gas stream in the
apparatus could be calculated.
An interval of about three minutes was allowed as a rest for the tube, as this made
it much more constant over a large number of readings. Iii the mean time, if
necessary, gas was forced back from the gas bag into the gasometer. Guided by the
previous electrometer deflection the potential of the outer cylinder was now changed,
and the whole process repeated. In this way a number of readings were taken, such
that the electrometer deflections ranged from some value down to near zero. These
were taken in such an order that at first, say, a descending series of readings was
obtained, and then immediately afterwards an ascending series. In this manner it is
possible to detect any uniform changes which are taking place in the intensity of the
rays, for in that case the two series of points would lie on curves of different
inclinations.
It was seen in § 5 that the time of passage of the ions from one cylinder to
the other could be varied by changing the velocity of the gas stream, and also by
changing the distance X. Both of these were employed in practice, and it was found
that the values of the velocity obtained diminished as the time increased ; but they
were practically the same for two different values of X if the velocity of the gas
stream was changed in the same ratio, i.e., if the time of passage of the ions was
the same.
J, S. TowNSEND* has recently observed that the rate of diffusion of the ions
depends upon the moisture in the gas. In these experiments the gases were used
both dry and saturated with aqueous vapour, and it was found that the velocity was
different in the two cases.
For saturating a gas with aqueous vapour it was forced, in passing between the
gasometer and the apparatus, to bubble through a water bottle and then to pass
through a long horizontal tube half filled with water. After the gas had been passed
several times back and forth between the gasometer and the gas bag, and before any
readings were taken, the water bottle was cut out so as to avoid any unsteadiness in
the pressure due to the bubbling.
For drying a gas the above arrangement was replaced by one in which the gas
had to pass through a long, horizontal glass tube, partly filled with concentrated
sulphuric acid, and then through a large volimie of calcium chloride. In order to
allow a suflBciently rapid stream with the small pressures used the calcium chloride
was placed in a large, wide bottle, the gas entering above and leaving by a protected
ftmnelshaped tube near the bottom. It thus had to traverse a considerable length of
calcium chloride, and on account of the large area of the bottle the velocity through
it was smaU,
♦ J, S. TowNSBND, 'Phil, Trans.,' A, vol. 193.
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208
MR. J. ZELENY ON THE VELOCITY OF THE IONS
§ 7. Moist Air.
The following is an example of a set of readings taken for the positive ions in air
saturated with aqueous vapour.
Letters refer to corresponding quantities in formula (9).
Temperature = 14*5° C X = 2'60 centims. a = '50 centims. b =
2 '555 centims.
Width of beam of rays = '20 centim. Barometer =75*4 centims.
Excess pressure inside gasometer =1*56 centims. of mercury.
„ „ in apparatus = '59 centim. of mercury.
20 cells = 42 "6 volts.
Table I. — Moist Air. Positive Ions.
Voltage of outer
cylinder. •
Electrometer deflection
Descent of gasometer
in 30 seconds.
in 40 seconds.
Cells.
Divisions.
Centims.
+ 10
145
677
+ 12
1055
679
+ 14
685
678
+ 16
295
672
+ 18
+ 19
12
670
683
7
+ 17
19
681
+ 15
525
678
+ 13
87
677
+ 11
128
676
In the middle of the observations the gasometer was refilled from the gas bag.
The sectional area of the gasometer was 2904 sq. centims., and the area between the
two cylinders was 19*73 sq. centims., so the average rate of descent of the gasometer
above indicates an average velocity in the apparatus of 25*2 centims. per second,
when corrected for the difference in pressure between the gasometer and the
apparatus.
The voltages and their corresponding deflections are represented graphically in
curve I. of fig, 5. The set of readings here given, and most of those which are to
follow as examples, have been selected from among the best obtained.
It is seen that the curve at first approaches the axis of abscissas in nearly a straight
line, but becomes convex when near to it. Had readings been taken for voltages
smaller than those used, that part of the curve would have been concave to the axis
of abscissas.
It has been explained in § 4 (3), why there is a nearly straight portion in the
curve, while the width of the beam of rays and the various causes tending to spread
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PRODUCED IN GASES BY RONTGEN RAYS.
209
the ions make the lower end of the curve approach the axis at a less rapid rate. It
was also shown that the point on the axis of abscissas, obtained by prolonging the
straight portion of the curve, would correspond to the voltage required to make an
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 /40
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— AQ ^
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i
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— 20
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5.
a
^
ion, starting from the surface of the outer cylinder in the middle of the beam of rays,
just reach the middle of the juncture in the inner cylinder. But with diffusion and
the other causes acting to produce a spreading of the ions, it is evident that the
inclination of the straight part itself is affected and the result changed. Corrections
for this error can only be made in conjunction with those of some other effects, and
that by experiment, by producing alterations in the amount of these effects, by
changes in the time of passage of the ions across the space between the cylinders.
The velocity obtained by the use of the voltage determined by the continuation of
the straight part of the curve, as shown in the figure, will be called the ionic velocity
for that determination, it being tmderstood that it is not implied thereby that the
velocity changes with the time, but that this is only a step towards the final result.
From the above curve, A is seen to be 177 cells, which is equal to 377 volts.
Using equation (9),
V = j — 5 — log<f~ f Tv = 5'118 ^^ — :^^ = 1'315 centims. per second,
The pressure in the apparatus is 76 centims. of mercury
VOL. cxcv.— A, 2 E
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210
MR. J. ZELENY ON THE VELOCITY OF THE IONS
From equation (10)
'*^ = i=ll = 'i*^^^"**
The following is a set of readings taken for negative ions in moist air. Unless
otherwise mentioned the values of a, 6, and the width of the beam of rays will
hereafter be taken the same as in the previous example.
Temperature = 14*4° C. X = 6*42 eentims. Barometer = 74*7 eentims.
Excess pressure inside gasometer = 1'54 eentims. of mercury.
„ „ in apparatus = '59 centim. of mercury.
8 cells =165 volts.
Table II. — Moist Air. Negative Ions.
Voltage of outer
CTlinder.
Electrometer deflection
Descent of gasometer
in 30 seconda.
in 40 seconds.
Cells.
Division*.
Centims.
4
128
605
5
• 685
595
54
45
594
6
175
592
56
325
589
52
50
590
6
18
602
54
445
599
5
675
596
44
95
590
7
2
590
The results are represented in Curve II. of fig. 5.
U = 22*1 centims. per second.
A = 127 volts.
221
= 1'39 centims. per second.
The pressure in the apparatus =75*3 centims.
The velocity reduced to 76 centims. pressure = 1'38 centims. per second.
T = ^ = 29 second.
The following is a summary of the results obtained for moist air for both the
positive and negative ions. Each result was obtained from a series of observa
tions as indicated by the above examples. The results are reduced to 76 centims.
pressure.
Letters refer to quantities in equations (9) and (10).
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PRODUCED IN GASES BY RONTGEN RAYS.
211
Tablk TTT. — Moiflt Air. Summary
of Besult£
).
Ionic velocity.
Reference
number.
X.
U.
A.
T.
Tempera
ture.
Gas
pressure.
Negative.
Positive.
1
433
206
+ 192
21
•c.
153
762
128
2
418
1085
+ 111
39
15
776
—
1225
3
418
111
101
•38
15
776
137
—
4
418
1073
 98
•39
143
765
135
—
6
418
1096
+ 1135
38
143
765
—
119
6
418
250
+ 234
•17
143
768
—
132
7
418
250
212
17
143
768
146
—
8
268
113
 1565
24
146
759
138
—
9
268
113
+ 175
24
146
759
—
123
10
268
220
+ 3215
12
14
761
—
132
11
268
221
291
12
14
761
146
—
12
841
1133
+ 626
75
144
760
—
110
13
841
1123
+ 603
75
144
760
—
113
14
841
1078
 52
78
144
760
126
—
15
841
1133
 541
74
144
760
127
—
16
841
118
 551
72
144
760
130
—
17
841
248
 1095
34
142
762
139
—
18
' 841
2473
+ 122
34
142
762
—
124
19
841
248
+ 125
34
142
762
—
122
20
841
248
107
34
142
762
142
—
21
642
1067
 657
60
135
758
130
—
22
642
107
+ 726
60
135
758
—
1175
23
642
2206
+ 140
29
144
748
—
1245
24
642
2206
+ 1415
•29
144
748
—
123
25
642
221
127
29
144
747
138
—
26
260
111
157
29
147
747
1375
—
27
260
111
+ 170
24
147
747
—
127
28
260
252
+ 377
•10
145
754
—
1315
. 29
260
252
341
•10
146
754
1466
—
30
260
126
178
21
15
755
139
—
31
260
109
157
24
15
755
137
—
32
260
131
183
•20
16
757
141
—
33
260
1316
178
•20
16
757
145
^
.7
.a
^ .3
T. In aeconds.
Fig. 6.
2 E 2
.£
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212
MR. J. ZELENY ON THE VELOCITY OF THE IONS
The results are represented iu fig. 6, where the velocities are represented as
ordinates and the corresponding values of T as abscissas. It is seen that the
velocities decrease with T, and nearly in a linear manner. Considerable variations
are observed among the individual results, but it is believed that they are not
greater than is to be expected from the nature and difficulties of the experiments.
In § 4 it was seen that some of the corrections which act to give too small a value
for the velocity diminish with T and disappear for T = 0. By drawing lines through
the points in fig. 6 and prolonging them to the axis of ordinates where T = 0, we
obtain the most probable values of the velocities. This gives for the negative ions
1*48 centims. per second, and for the positive ions 1*34 centims. per second.
In § 4 (15) it was stated that a correction of 2 per cent, would be made for dis
turbances not corrected by the above method. This gives for the final results for
moist air the velocity in an electric field of 1 volt per centim. for the negative
ions =1*51 centims. per second, and for the positive ions = 1'37 centims. per second
at a temperature of about 14° C, and a pressure of 76 centims. of mercury.
§8. Dry Air.
The following set of readings was taken for the positive ions in dry air : —
Temperature = 13'8° C. X = 2'60 centims. Barometer = 761 centims.
Excess pressure in gasometer = 1 "6 centims, of mercury.
„ „ apparatus = '45 „ „
14 cells = 290 volts.
Table IV. — Dry Air. Positive Ions.
Voltage of outer
cylinder.
Electrometer deflection
Descent of gasometer
in 30 seconds.
in 40 seconds.
Cells.
Divi«ioiia
Centims. 1
+ 8
117
429
+ 10
60
428 !
+ 12
22
426
+ 14
7
426
+ 11
40
422
+ 9
94
42.5
+ 7
153
425
+ 10
62
421 !
+ 8
123
423
These results are represented graphically in Curve I. of fig. 7.
The corrected value of U is 15 '9 centims. per second.
A = 248 volts.
159
Sov = 5118
260 X 24'8
= 1*26 centims. per second, and when reduced to
76 centims. pressure this becomes 1'27 centims. per second.
rv 260 ,^ J
T = — = 16 second.
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PRODUCED IN GASES BY RONTGEN RAYS.
The following set of readings was taken for the negative ions in dry air : —
Temperature = 15'8° C. X = 2*60 centims. Barometer = 76 centims.
Excess pressure in gasometer = TO centim.
„ „ „ apparatus = '13 centim.
7 cells =145 volts.
213
Fig. 7.
Table V. — Diy Air. Negative Ions.
Voltage of outer
cylinder.
Electrometer deflection
Descent of gasometer
in 30 seconds.
in 40 seconds.
Celli.
DiTisions.
Centims.
2
1415
207
3
87
205
4
285
208
6
65
205
44
18
208
34
63
205
24
116
205
The results are shown graphically in Curve IL of fig. 7.
U corrected for pressure = 7 '64 centims. per second.
A = 921 volts.
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214
V = 5118
MR J. ZELENY ON THE VELOCITY OF THE IONS
764
260 X 921
— = 1*63 centima per second.
T =  If? = 34 second.
A summary of the results obtained for dry air for both the positive and the
negative ions is given in Table VI.
Table VI. — Dry Air. Summary of Results.
Ionic velocity.
Beference
X.
U.
A.
T
Tempera
Gas
number.
J •
ture.
pressure.
Negative.
Positive.
I
260
764
 921
34
°c.
158
762
163
2
260
764
+ 120
34
158
762
125
3
260
764
+ 122
34
163
758
123
4
260
757
 931
34
163
758
160
5
260
728
 915
36
122
766
158
6
260
723
+ 1215
36
122
766
M8
7
260
158
 183
16
138
766
171
8
260
159
+ 248
16
138
766
—
127
9
260
159
18.1
16
138
7«6
174
10
260
159
183
16
.138
766
1725
11
260
166
 1865
16
146
758
175
12
260
163
 1822
16
14
768
178
13
260
155
+ 248
17
125
767
125
14
260
155
+ 237
17
107
776
131
15
260
156
181
17
107
776
172
16
515
865
+ 776
60
114
773
—
113
17
515
862
 59
60
114
773
147
18
515
156
 931
33
114
776
167
19
515
157
+ 126
33
114
776
126
20
515
858
+ 764
•60
117
775
_
114
21
515
8^58
 588
60
117
775
148
At No. 7 the drying apparatus was changed, and at No. 12 the Crookes tube
was replaced by a new one.
The results are represented in fig. 8, excluding the points marked by squares.
The final values thus obtained for dry air when the 2 per cent, correction men
tioned in § 4 (15) has been added, give the velocity of the negative ions = 1 '87 centims.
per second, and of the positive ions = 1*36 centims. per second.
The temperature varied several degrees between the difierent observations, but
was on the average about 13° C.
Most of the tests to which the method used in these experiments was subjected
by changes of experimental conditions, were tried with dry air. Among these was
tried the effect of changes in the intensity of the rays. By interposing aluminium
plates the rays were diminished so that the conductivities produced by them changed
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PRODUCED IN GASES BY RONTGEN RAYS.
215
in the ratio of three to one, but no noticeable change in the result could be observed.
During the course of all of the experiments the rays were not of the same intensity,
for the Crookes' tube had to be replaced sevei'al times, but in aU cases without any
marked effect upon the values obtained. It must be said, however, that rays of great
intensity were never employed, the aim being always to have them as weak as
possible for reasons previously stated.
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The most severe teat to which the method was subjected was a change in the
dimensions of the inner cylinder. In the above experiments the diameter of this
cylinder was 1 centim., and it was now exchanged for one having a diameter of
2 '8 centims. The distance between the inner and the outer cylinders was thus
diminished to nearly onehalf of its former value. The electric field between the two
became much more uniform, and the gas velocities for different points of a cross
section now varied in a different manner. In order to keep the other quantities the
same, the small distance between the two cylinders necessitated the use of voltages
only about onequarter as large as those used in the former arrangement. This
increased the difficulty of the measurements and also some of the corrections which
must be applied to get the final result. The density of the free charges in the gas
was greater because the ions moved slower, being in a weaker field, and the same fall
of potential at the electrodes was a larger percentage of the total voltage. The
width of the beam of rays used was '3 centim.
The following is a summary of the results obtained : —
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216
MR. J. ZELENY ON THE VELOCITY OF THE IONS
Table VII.— Dry
■ Air. Summary
for Large
Inner Cylinder.
Ionic velocity.
Beference
number.
X.
U.
A.
T,
Tempera
ture.
Gas
pressure.
Negative.
Positive.
1
54
107
190
50
"C.
irs
75^9
143
_
2
54
126
223
•43
124
759
143
—
3
54
127
+ 284
•43
124
759
—
113
4
54
104
+ 235
52
12^4
75^9
—
110
5
377
1155
+ 375
33
15
75^8
112
6
377
1155
270
•33
15
75^8
1^555
7
377
1165
280
33
15
75^8
151
8
511
139
264
37
154
76^8
1^43
9
511
1395
+ 339
•37
15^4
76^8
1115
10
326
138
406
•24
15^4
76^8
157
11
636
883
148
•72
16
77^2
1315
12
635
883
+ 180
72
16
77^2
—
108
13
636
1405
+ 280
•45
15^8
76^0
—
109
U
635
142
220
•45
15^8
760
140
15
263
149
47
•18
15^8
760
165
16
263
77
+ 35
•34
163
75^9
—
M4
17
263
704
r25
•37
163
75^9
1465
18
263
139
432
•19
163
759
167
19
263
1406
+ 594
•19
163
75^9
123
The results are represented in fig. 9.
y^
^
^
l^
y^
J
^^^
aC
Ve
v^
^
O
^
^
o
^
^
^
o
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^
^
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i
^
m
fe.
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^
^
^^
^
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—
►—
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k
iH
r^
•7
fr
€
f
s
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^
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•<s
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. ./
\
fl
r in seconds.
Fig. 9,
hS
1.7
i.5
$
.^
1.3
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J
ht
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PRODUCED IN GASES BY RONTGEN RAYS.
217
It is seen that in this case the values of the velocity change less rapidly as the
values of T become large ; but for the smaller values of T the change is more rapid
than it was when the smaller inner cylinder was used. The points on the curves are
not advantageously distributed, and so do not allow of a very accurate projection of
the lines to T = ; but from those drawn it is seen that the results are but slightly
smaller than those obtained with the smaller inner cylinder. This is considered a
good agreement even if it is left out of account that an additive correction is still to
be made.
An alteration which was tried to test the eflfect of surface ionization was a change
in the material of the inner surface of the outer cylinder. The aluminium part DD'
(fig. 2) of the outer cylinder was coated on its inner surface with a layer of tinfoil.
The rays in penetrating the cylinder now had a tin instead of an aluminium surface
in contact with the air. J. Perrin has shown (see § 4 (10) ) that what he calls the
coefficient of the increased ionization at a metal surface is for tin in air '6 as against
•0 for almninium in air. The effect varies with the state of the surface. In these
experiments the aluminium surface used was an ordinary unpolished surface, while
the tin surface used was that of bright tinfoil. It was thought that if an increase
of the ionization near the metal surface has any marked effect upon the value of the
velocity obtained, the difference should be observed by this new arrangement.
The results obtained are given in Table VIII. , dry air being used as before. The
smaller inner cylinder having a diameter of 1 centim. was used.
Table VIII.
— Dry Air. Summary for Tin Surfiice.
Ionic velocity.
Beference
number.
X.
U.
A.
T.
Tempera
ture.
Q&8
pressure.
Negative.
Positive.
1
622
856
 606
•6
"C.
139
773
142
2
622
861
+ 782
•6
139
773
—
1105
3
622
182
+ 152
•29
144
77^4
—
120
4
522
182
108
•29
144
77^4
1^68
5
262
173
196
•15
»
772
176
6
262
185
212
•14
165
77^2
173
—
7
262
185
+ 294
•14
16^5
77^2
125
The points are plotted as squares on the curves in fig. 8, which represent corre
sponding values when the aluminium surface was used. It is seen that the points
for the negative ions agree very well with the curve. The points for the positive
ions are 2 to 3 per cent, below the values for the aluminium surface. Taking both
results into consideration it was concluded, if the addition of a tin surface changed
the values of the velocities by but such a small amount, that originally when the
VOL. OXCV. — ^A. 2 F
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218
MR. J. ZELENY ON THE VELOCITY OF THE IONS
aluminium surface was used, the eflfect of the surface ionization could not have been
sufficient to produce any marked error in the results.
The surface ionization also varies with the nature of the gas, but the values
obtained by J. Permn for aluminium with the gases used in these experiments were
in all cases much less than for tin in air.
§ 9. Oxygen.
The gas used in these experiments was the commercial oxygen obtained from a
cylinder, which contained about 5 per cent, of impurities, mostly nitrogen. Since
the size and nature of the apparatus prevented the employment of the most pure
gases, it seemed advisable to use the cylinder gas. The density was changed but
little by the presence of the impurities, and, so far as known, the velocity should
therefore be but slightly aflfected. The drying of the gas and its saturation with
aqueous vapour were carried out in the same manner as with air. The following set
of readings was taken for the negative ions in oxygen saturated with aqueous
vapour : —
Temperature = 173° C. X :
Excess pressure in gasometer
„ „ apparatus
8 cells = 163 volts.
5*01 centims.
1*54 centims.
•55 centim.
Barometer = 76*4 centims.
Table IX. — Moist Oxygen. Negative Ions.
Voltage of outer
Electrometer deflection
Descent of gasometer
cylinder.
in 30 seconds.
DivisionB.
in 40 seconds.
Cells.
Centims.
3
1945
606
4
157
511
5
1065
502
6
41
511
66
17
507
56
67
508
52
935
509
46
1255
503
The results are shown in Curve I. of fig. 10.
The corrected value of U = 1883 centims. per second
A = 1355 volts.
1883
V = 5118
= 1*413 centims. per second, and when reduced to
T =
501 X 1355
76 centims. pressure this becomes 1'43 centims. per second,
501
1883
= '27 second,
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PRODUCED
IN
GASES
BY RONTGEN RAYS.
J
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219
The following is a set of readings taken for the positive ions in oxygen saturated
with aqueous vapour : —
Temperature = 15*6° C. X = 5'01 centims. Barometer = 765 centims.
Excess pressure in gasometer = '44 centims.
„ „ apparatus = '16 „
6 ceUs = 1235 volts.
Table X. — Moist Oxygen. Positive Ions.
Voltage of oater
Electrometer deflection
Descent of gasometer
cylinder.
in 30 seconds.
in 40 seconds.
Cells.
Diviaions.
CentiniB.
+ 2
US
230
+ S
856
231
+ 36
206
229
+ 32
41
229
+ 28
736
227
+ 24
112
225
+ 22
127
226
The results are represented by Curve 11. of fig. 10.
2 F 2
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220
MR. J. ZELENY ON THE VELOCITY OF THE IONS
The corrected value of U = 8 "42 centims. per second.
A = 742 volts.
V = 5118
842
= 1'16 centims. per second, and when reduced to
501 X 742
76 centims. pressure this becomes 1'17 centims. per second.
T = —r^ = '6 second.
o'4^
A summary of the results thus obtained for moist oxygen for both the positive and
the negative ions is given in Table XI.
Table XI. — Moist Oxygen. Summary of Results.
Beference
number.
X.
U.
A.
T.
Tempera
ture.
Gas
pressure.
Ionic velocity.
Negative.
Positive.
1
2
3
4
5
6
7
8
9
10
689
689
689
689
501
501
501
501
303
303
941
938
168
168
842
868
188
186
205
206
+ 604
 538
 880
+ 105
+ 742
 669
 1355
+ 157
+ 273
 2425
•73
•73
•41
•41
•6
•58
•27
•27
•15
•15
"C.
15^4
154
156
155
156
156
173
173
154
154
768
76^8
773
773
76^7
76^7
769
769
769
769
131
144
134
143
1^46
117
fil
117
123
129
The results are shown graphically in fig. 11.
.6
.5
T in seconds.
Fig. 11.
.3
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PRODUCED IN GASES BY RONTGEN RAYS.
221
The correction mentioned in § 4 (15), which is to be applied to the values indicated
at T = has in this case been reduced to 1 per cent., 1 per cent, being allowed for
an increase in the velocity due to a diminution of density caused by the impurities in
the gas. The corrected value thus obtained for the velocity in moist oxygen is for
the negative ions = 1'52 centims. per second, and for the positive ions = 1*29
centims. per second, at a pressure of 76 centims. and at a temperature of about
16° C.
The following is a summary of the results obtained for the positive and negative
ions in dry oxygen : —
Table XII. — Dry Oxygen. Summary of Results.
Beference
number.
X.
U.
A.
T.
Tempera
ture.
Gas
pressure.
Ionic velocity.
Negative.
Positive.
1
2
3
4
6
6
7
8
9
10
11
12
13
14
273
273
273
273
273
389
389
389
389
389
689
689
689
689
139
138
133
133
133
1665
177
169
797
846
892
907
1735
174
+ 202
 1537
+ 195
147
+ 195
136
+ 177
134
 682
+ 945
+ 564
 463
 806
+ 1033
20
20
21
•21
21
23
22
23
•49
•46
77
76
40
40
"0.
203
203
194
194
194
16
156
166
16
16
152
152
158
158
773
773
774
774
774
78
78
78
768
76^8
771
771
771
771
171
172
166
171
156
fsi
163
131
131
130
135
119
119
127
The results are represented graphically in fig. 12.
When, as in the case of moist oxygen, a 1 per cent, additive correction is applied
to the values indicated in the figure by T = 0, the final result for the velocity in dry
oxygen is for the negative ions = 1'80 centims. per second, and for the positive ions
= 1'36 centims. per second for a pressure of 76 centims. and a temperature of about
17^ C.
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222
MR. J. ZELENY ON THE VELOCITY OF THE IONS
— 1
'
—
—
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Ol^
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—
^
/•/
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f
A
5
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,*
4
.J
.£
J
r
I
*
I
in seconds.
Fig. 12.
§ 10. Carbonic Acid.
The gas used was taken from a cylinder of liquid carbonic acid. The small amount
of impurities in this does not produce any marked change in the density of the gas,
and is assumed to be without noticeable effect upon the ionic velocities. As examples
of the readings taken, the following two sets are given for carbonic acid gas saturated
with aqueous vapour : —
Temperature = 16'3° C. X = 302 centims.
Excess pressure in gasometer = '44 centim.
„ „ „ apparatus = '21 „
10 cells = 206 volts.
Barometer = 75*4 centims.
Table XIII, — Moist Carbonic Acid.
Negative Ions.
Voltage of outer
cylinder.
Electrometer deflection
Descent of gasometer
in 30 seconds.
in 40 seconds.
Celb.
DiTisions.
Centims.
 6
136
234
 7
1065
233
 8
76
233
 9
415
2.30
10
175
233
 94
28
229
 84
57
228
 74
915
227
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PRODUCED IN GASES BY RONTGEN RAYS.
The results are shown in Curve I. of fig. 13.
2'23
lao
VolCcL^e in ceils.
Fig. 13.
The corrected value of U = 8*52 centims. per second.
A = 21'1 volts.
'302 n4 ~ ^^^ centim. per second, which, reduced to 76
V = 5118:
centims. pressure, becomes '679 centim. per second.
m 302 „^ J
T = T7^ = 36 second.
Table XIV. — Moist Carbonic Acid. Positive Ions.
Voltage of outer
cylinder.
Electrometer deflection
Descent of gasometer
in 30 seconds.
in 40 seconds.
OelU.
Divisions.
Centims.
+ 2
1765
232
+ 3
170
231
+ 6
106
233
+ 7
775
232
1 +8
45
232
+ 9
17
232
+ 84
28
226
+ 74
62
225
+ 64
i
975
227
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224
MR. J. ZELEITS ON THE VELOCITY OF THE IONS
These results are represented in curve II. of fig. 13.
The corrected value of U = 8 "48 centims. per second.
A = 1916 volts.
848
.= "749 centim. per second, which reduced to 76 centims.
"  ^'^^^ 302 X 1916
pressure becomes "745 centim. per second.
3*02
T = r^ = 36 second.
o.4o
A summary of the results for moist carbonic acid is given in Table XV.
Table XV. — Moist Carbonic Acid. Summary of Results.
Reference
number.
X.
U.
A.
T.
Tempera
ture.
Gas
pressure.
Ionic velocity.
Negative.
Positive.
1
2
3
4
5
6
7
8
9
10
11
302
302
302
302
607
6.07
607
607
607
607
607
848
852
166
1675
974
997
194
181
863
1305
129
+ 1916
211
392
+ 359
123
+ 11
+ 215
2214
+ 995
+ 141
 1585
36
36
•18
■18
62
61
•31
•33
•70
•47
•47
°C.
163
163
166
166
16^9
169
17^1
171
17^1
176
176
756
756
762
76^2
75
75
755
755
75
75^1
75^1
•679
•717
•658
•685
•678
•745
•791
•755
756
•722
•772
^^
^
P
73/
f/V
e.
—
"^
■^
.
»
. .
—
*
X
^^
^
—
— ■
"*"
X
^^
^
— ■
,
—
^
' —
"^
ft
^^
— 
"^
a—
....
—
—
Ni
J?^
th
e.
"^
•1
• i
7
X
\
.i
1
4
^
.«
./
»
mi
f
^i
i
4^
J
T in seconds.
Fig. 14.
The results are represented in fig. 14, from which it is seen that the values
corresponding to T = 0, when corrected similarly to those of moist oxygen, give as
the velocity in moist carbonic acid, for the negative ions, 75 centim. per second, and
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PRODUCED IN GASES BY RONTGEN RAYS.
225
for the positive ions, '825 centim. per second, for a pressure of 76 centims. and a
temperature of about 17° C.
A summary of the results obtained for dry carbonic acid is given in Table XVI.
Table XVI. — Dry Carbonic Acid. Summary of Results.
Beference
number.
X.
U.
A.
T.
Tempera
ture.
Gas
pressure.
Ionic Velocity.
Negative.
Positive.
1
2
3
4
5
6
7
8
9
10
11
12
607
607
607
607
308
308
308
308
601
601
601
601
861
863
167
171
173
173
825
853
863
853
128
1285
 915
+ 959
+ 1907
 1805
362
+ 384
+ 1896
1818
 953
+ 964
+ 1476
 1404
•71
•71
•36
•36
•18
•18
•37
•36
•70
•71
•47
•47
c.
175
175
175
175
183
183
172
172
173
173
175
175
754
754
758
758
758
758
757
757
757
75^7
759
759
787
•796
•793
•781
•770
•777
•752
•737
•747
•725
•753
•738
The results are represented in fig. 15.
The velocities appear to vary but little with T.
The values for the positive velocity being comparatively large for the highest value
of T, make it difficult to draw the line through the positive points, and the inclination
of the one through the negative points has been used as a guide for drawing the one
shown. The value thus found, when corrected, gives the velocity in dry carbonic acid
for the negative ions as '81 centim. per second, and for the positive ions as 76 centim.
per second for a pressure of 76 centims. and a temperature of 17*5° C.
i
N€
S/K
?.
=sss
—
^
—
<t
=
°"
_a
^^^
^
—
=
X
F\
^Si
Wi
f.
y%
J
. d
f —
.€
\
4
1
,4
^
•%
1
\ —
•i
t
s
VOL. CXCV. — A.
T in seconds.
Fig. 15.
2 G
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226
MR. J. ZELENY ON THE VELOCITY OF THE IONS
§ 11. Hydrogen.
The gas was prepared from pure zinc and hydrochloric acid, and bubbled through
three bottles of strong caustic potash and potassium permanganate to free it from the
acid and other impurities. Great difl&culty was experienced in maintaining the gas
sufficiently pure on standing, because of the large surface of rubber exposed in the
gas bag and in the connecting tubes of the apparatus. The density of hydrogen
being so small compared to that of air, a small amount of the latter produces a large
change in the density of the gas, and it was found that the ionic velocities were
greatly affected thereby.
The following plan was finally adopted as the most practicable under the circum
stances : —
The forenoon of a day was spent in the preparation of fresh hydrogen, this length
of time being required to generate the large quantity necessary for use and for
washing the more impure hydrogen out of the apparatus. Beginning early in the
afternoon, readings were taken as rapidly as possible until after midnight, thus giving
about eleven hours of continuous observations. The density of the gas was then
determined by weighing a 600 cubic centims. flask filled first with dry air and then
with dry gas from the gasometer. Since 1 per cent, of air in the gas made a difference
of over 6 milligrams in the weight, this permitted a sufficiently accurate determination
of the amount of the air impurity. A test was made by the eudiometer method,
which showed that the impurity was practically all air.
The width of the beam of rays used was '3 centim., as the conductivity was much
less with the hydrogen than in the other cases.
The following is a set of readings taken for the negative ions in dry hydrogen : —
Temperature = 20° C. X = 2*95 centims. Barometer = 76*15 centims.
Excess pressure in gasometer = '90 centim.
„ ,. apparatus = 'SG centim.
5 cells = 10*5 volts.
Table XVII.* — Dry Hydrogen. Negative Ions.
Voltage of outer
cylinder.
Electrometer deflection
Descent of gasometer
in 30 seconds.
in 40 seconds.
Cells.
Diviiiouii.
Centinis.
26
125
959
3
97
952
34
65
931
38
38
944
36
6
946
32
. 88
950
28
112
938
24
142
940
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PRODUCED IN GASES BY EONTGEN RAYS.
227
The results are shown graphically in curve II. of fig. 16.
The corrected value of 17= 35*0 centims. per second.
A = 904 volts.
V = 5118
35
295 X 904
pressure becomes 676 centims. per second.
9Q5
T = ^  = 084 second.
The gas in this case contained 3 '4 per cent, of air.
= 672 centims. per second, which reduced to 76 centims.
L
f
n
J
/
lA
7
f
J
F^
t
/
lO
7
y
%
L
/
A
1
/
^
t
7
^»^
'4
M
/
O
t
t
f
J
7
7
A
4± ^
(
7
i^"'
7
f
J
t
/
lV
: : ( ^. i
u
4t.
VoLta^ in ceUs.
Fig. 16.
The following set of readings was taken for the positive ions in hydrogen saturated
with aqueous vapour : —
Temperature = 20° C. X = 2*95 centims. Barometer = 767 centima
Excess pressure in gasometer = '78 centim.
„ „ apparatus = '35 centim.
9 cells =185 volts.
2 G 2
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228
MR. J. ZELENY ON THE VELOCITY OF THE IONS
Table XVIII. — Moist Hydrogen. Positive Ions.
Voltage of outer
Electrometer deflection
Descent of gasometer
cylinder.
in 30 seconds.
in 40 seconds.
Cell..
DiTisiona.
Centims.
+ 66
525
1038
+ 6
775
1023
+ 54
1025
1018
+ 46
14
1038
+ 54
1075
1025
+ 6
775
1018
+ 66
5
1016
The results are shown in curve I. of fig. 16.
The corrected value of U = 43 '3 centims. per second.
A = 159 volts.
433
i; = 5118
<^n^ i crn = 473 centims. per second, which reduced to 76 centims.
295 X 159 ^
pressure, = 4'80 centims. per second.
2*95
T = ^ = '068 second.
Besides the water vapour, the gas in this case contained 1 "5 per cent, of air.
A summary of the results obtained with dry hydrogen containing 3*4 per cent, of
air is given in Table XIX. On account of the smaller electrometer readings the
Table XIX. — Dry Hydrogen. Summary of Results.
Beference
number.
X.
U.
A.
T.
Tempera
ture.
Gas
pressure.
Ionic velocity.
Negative.
Positive.
1
2
3
4
5
fi
7
8
9
10
11
12
13
14
15
16
295
295
295
295
295
295
295
295
295
295
295
295
295
295
295
295
212
216
218
219
216
214
214
214
354
353
350
350
349
347
257
257
+ 665
+ 773
+ 714
+ 737
 579
 575
 585
+ 714
+ 1035
+ 1079
 903
 924
 890
+ 1110
+ 808
+ 790
083
083
084
•084
084
085
115
115
•u.
214
214
214
214
214
214
214
214
20
20
20
20
20
20
20
20
763
763 '
763 '
763
763
763
763
763
765
765
765
765
765
765
76 4
764
652
649
639
677
6eo
684
556
515
533
518
524
597
571
545
555
567
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PKODUCED IN GASES BY RONTGEN RAYS.
229
determinations for hydrogen are less accurate, and so results were obtained for small
values of T, only because of their greater importance, and in order to expedite the
readings.
The results are represented by I. and II. of fig. 17.
The lines projected to T = indicate for the uncorrected velocity of the negative
ions 7 '3 centims. per second, and for the positive ions 6*2 centims. per second when
under a pressure of 76 centims., and at a temperature of about 20° C. These values
are for dry hydrogen containing 3*4 per cent, of air. The correction for the presence
of the air can be found approximately by finding the value of the velocity in a gas
having a larger percentage of air.
z
Ni
5^
th
e.
y
X'
/.
.^
y
7.
O J
/
o
y
^
i
i
°«
iCi
ve.
y
«0
— 3r
y
X'
« 2j,
X
X
^
• t^
M
y
X
f
^
»
n.
•5
w
^
o
X
^
^
''^
^
^
^
^*
X
n,
^
»l
\
•
♦2
x>
•
T in seconds.
Fig. 17.
./
The following are a number of results obtained with dry hydrogen which contained
14*4 per cent, of air : —
Table XX. — Dry Hydrogen with 14*4 per cent, of Air.
Reference
number.
X.
U.
A.
T.
Tempera
ture.
Gas
pr/)ssure.
Ionic velocity.
Nogative.
Positive.
1
2
3
4
295
295
295
295
204
202
186
186
+ 962
828
759
+ 890
•15
•15
•16
•16
"C.
212
212
■ 22
22
763
763
764
764
425
427
370
365
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230
MR. J. ZELENY ON THE VELOCITY OF THE IONS
These results are represented at III. in fig. 17.
By finding the difference between these points and the values in the curves above
them corresponding to the same value of T, the diminution in the velocity is obtained
that is produced by the addition of 14*4 — 3 4 = 11 per cent, of air. Assuming that
up to this point the diminution in the ionic velocity is proportional to the amount of
air present in the gas, the velocity in pure hydrogen is found by adding to the value
34
obtained when 3 '4 per cent, of air was present tj part of the diminution observed as
due to 11 per cent, of air. From the above results this correction is found to be '65
for the negative ions and '50 for the positive ions. Disregarding any minor correc
tions, the final result for pure dry hydrogen is thus found to be 7 '9 5 centims. per
second for the negative ions, and 6*70 centims. per second for the positive ions at a
pressure of 76 centims., and at a temperature of about 20° C.
A summary of the results obtained with hydrogen saturated with aqueous vaj)our,
and containing 1 '5 per cent, of air, is given in Table XXI.
. Table XXI.
—Moist Hydrogen. Summary of Results.
Ionic velocity.
Reference
number.
X.
U.
A.
T.
Tempera
ture.
Gas
pressure.
Negative.
Positive. I
1
295
439
U6
•067
20
771
526
2
295
437
150
•067 1
20
771
510
—
3
295
437
+ 155
•067
20
771
—
497
4
295
433
+ 159
•068
20
771
—
480
5
295
431
150
•069
20
771
503
—
6
295
238
 843
•12
19^8
.76^9
4^98
—
7
295
235
 876
•13
19^8
76^9
472
—
8
295
234
 864
•13
19^8
76^9
4^77
—
9
295
233
+ 877
•13
19^8
769
468
10
295 233
+ 869
•13
198
76^9
473
11
295 I 342
+ 133
•087
19^8
77
453
12
295
341
+ 138
•087
19^8
77
437 i
13
295
34
124
•087
198
77
482
—
14
298
198
 874
•15
204
767
393
—
15
298
197
+ 897
•15
207
76^9
■~~"
382
air
The results 1 to 13 are represented by IV. of fig. 17.
The resuJt^ 14 and 15 were obtained with moist hydrogen containing 8 per cent, of
ThesQ itwp were selected out of a number of results of which they represent
about the ,averag€ values. They are shown by V. of fig. 17, and by means of them
the correction for the air present in the above experiments was made in the same
manner as with dry hydrogen. The points IV. in the figure are so scattered that the
inclination of the lines drawn through them had to be estimated mainly by
comparison with those for dry hydrogen, remembering that with the smaller
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PRODUCED IN GASES BY RONTGEN RAYS.
231
velocities here obtained the inclination would be somewhat less. The final values
thus obtained for hydrogen saturated with aqueous vapour when corrected for the air
present give for the velocity of the negative ions 5 '6 centims. per ^second, and for the
positive ions 5*3 centims. per second at a pressure of 76 centims., and a temperature
of 20° C.
§ 12. Eemarks on the Experiments.
The changes in the values obtained for the velocity with changes of T are
observed to be greater for those cases where the ionic velocities are higher.
With dry and moist carbonic acid, however, the inclination of the ciu*ves is some
what different for nearly equal values of the velocities. In some instances, where
the set of points for either the positive or the negative ions did not allow of a
suflSciently accurate estimate of the inclination of the line to be drawn through them,
the line through the other set of points was used as a guida
The presence of water vapour diminished the velocity of the negative ions in all of
the gases, while in carbonic acid the velocity of the positive ions was at the same
time considerably increased. It seems most probable that these changes are due to
some effect upon the size of the ions, and it is possible that a few molecules of the
aqueous vapour collect upon the negative ions. It is interesting to note in this con
nection the recent results of C. T. R. Wilson,* showing that in supersaturated air
the water condenses more readily upon the negatively charged ions.
While in most cases the readings indicate a greater accuracy, it is believed that
the maximum error in any determination is less than five per cent. For convenience,
all of the values obtained are here collected in one table, the results being given in
centims. per second both for a field of one volt per centim. and for a field of one
electrostatic unit per centim.
Table XXII. — Ionic Velocities.
Gas.
Velocities in centims.
per second in a field of
1 volt per centim.
Velocities in centims.
per second in a field of
1 E.S.U. per centim.
Eatio
of
Negative
to
Positive.
Tempera
»ture.
Positive
ions.
Negative
ions.
Positive
ions.
Negative
ions.
Air, dry
„ .moist
Oxygen, dry ....
„ moist . . .
Carbonic acid, dry . .
„ „ moist .
Hydrogen, dry . . .
„ moist . .
136
137
136
129
•76
•82
670
530
187
r5i
180
1^52
•81
•75
7^95
560
408
411
408
387
228
246
2010
1590
561
453
540
456
243
225
2385
1680
1375
MO
132
1^18
107
•915
119
105
"C.
135
14
17
16
175
17
20
20
* C. T. R. Wilson, *Phil. Trans.,' A, vol. 193, p. 289, 1899.
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232 MK. J. ZELENY ON THE VELOCITY OF THE IONS
It is seen that the value of the velocity is greater for the negative ions in all cases
except for moist carbonic acid. In comparing the values for the diflferent gases, the
temperature at which the observations were taken must be taken into consideration.
At the time the writer* determined the ratio of these velocities, the influence of
moisture being unknown, the gases used were not dried, and so the values obtained
were between those given above for the dry and the moist gases. Of the gases used
in the former experiments, which were not used in these, the ammonia gas used had
been passed through two long tubes of calcium oxide, the acetylene gas had been
passed through a long tube of calcium carbide and the nitrogen monoxide was used
directly from a cylinder.
The results obtained by E. RuTHERFORDt for the sum of the velocities of the ions
produced by Rontgen rays are for : —
Air = 3'2 centims. per second.
Oxygen =2*8 centims. per second.
Carbonic acid = 2*15 centims. per second.
Hydrogen = 10*4 centims. per second.
It is not stated whether the gases were dried, but the value for air agrees with
that given above for the sum of the velocities in dry air, while the values for oxygen
and hydrogen agree with the values for the moist gases. The value for carbonic acid
is nearly 40 per cent, larger than that obtained here. It is of interest to compare
the velocities of the ions produced by Rontgen rays with those of the ions produced
by the action of ultraviolet light and in the discharge from points, as they show a
close similarity.
For conduction produced by ultraviolet light, E. Rutherford:; obtained with
dry gases for the velocity of the negative ions in —
Air = 1'4 centims. per second.
Hydrogen = 3*9 centims. per second.
Carbonic acid = '78 centim. per second.
The value for carbonic acid is quite near to that obtained above, but the other two
are considerably smaller.
A. P. Chattock§ obtained for the velocities of the ions in dry air in the case of
discharge from points —
413 centims. per second for the positive ions, and
540 „ „ „ negative ions for a field of one electrostatic unit.
* J. Zeleny, *Phil. Mag.,' July, 1898.
t E. Rutherford, ' Phil. Mag.,' November, 1897.
t E. RuTHERiX)iU), ' Proc. Camb. Phil. Soc.,' vol. 9, Pt. VIII.
§ A. P. Chattock, *Phil. Mag.,' November, 1899.
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PRODUCED IN GASES BY RONTGEN RAYS.
233
These values are nearly the same as those obtained above for the ions produced by
Rontgen rays.
J. S. TowNSEND* has shown that from the ionic velocity in a gas and the coeffi
cient of diffusion of the ions in the gas, the value of Ne can be obtained, N being
the niunber of molecules in a cubic centim. of the gas, and e the charge carried by
an ion. By comparing this value with that obtained from the electrolysis of liquids,
the relation between the charges on the ions in the two cases can be determined.
Using the values of the ionic velocities (v) given in Table XXII., and the corre
sponding coefficients of diffiision (K) from the tables given by J. S. Townsekd, the
^ V 1 ft® 1)
values of Ne are obtained from the equation Ne = — = for the positive and the
negative ions in both dry and moist gases.
The results are given in the following table : —
Table XXIIL— Values of Ne X lO'i^.
Gas.
Moist gas.
Dry gas.
Positive
ions.
Negative
ions.
Positive
ions.
Negative
ions.
Air
128
134
124
101
129
127
M8
•87
146
163
1'63
131
136
125
Oxvfiren
Hvdrofifen
Carbonic acid
•99
•93
The corresponding value of Ne obtained for hydrogen from the electrolysis of
liquids is 1*23 X 10^^ at a pressure of 76 centims. of mercury, and a temperatiu*e
ofl5°C.
The values of Ne in the table for the positive and the negative ions in moist air,
oxygen and hydrogen are perhaps in sufl&cient agreement to justify the statement
that the charges carried by the positive and negative ions are the same, and that the
value is also the same for the three gases, and corresponds to the charge carried by
the hydrogen ion in the electrolysis of liquids.
The values of Ne for the negative ions in the same three gases when dry are not
far from those in the moist gases, but the results for the positive ions are consider
ably larger. It seems very improbable, however, that the charges carried by the ions
are different in the moist and dry gases, since most likely the moisture does not in
fluence the act of the ionization itself, but either affects the ions after they are formed
during the production of clusters of molecules around them, or changes the resistance
VOL. CXCV. — Jl,
* J. S. TowNSEND, *Phil. Trans.,' A, vol. 193, p. 152.
2 H
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234 VELOCITY OP IONS IN GASES.
to their motion. So if the charges are equal in the moist gases, they should be equal
in the dry gases also.
The values of Ne for carbonic acid are all less than that obtained for hydrogen by
electrolysis, and so indicate a smaller charge on the ions ; but from analogy with
liquids we should expect that if the charges vary at all, it would be in the ratio of
one to two or more, unless it is possible to have a charge smaller than that carried by
hydrogen in electrolysis.
The writer cannot account for the differences in the values of Ne by supposing
them due to errors in the ionic velocities obtained, since that would mean the pre
sence in the experiments of some error which in some cases influenced the results
for the positive ions alone, in other cases had an effect upon the values of both of
the ions, and in still other cases was without effect.
The experiments described in this paper were performed at the Cavendish Labo
ratory, Cambridge, and I desire to express here my thanks to Professor J. J. Thom
son for the encouragement and valuable suggestions given in the course of the
investigation.
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[ 235 ]
VI. Undergrmind Temperature at Oxford in tJie Year 1899, as determined by
Jive Platinumresistance Thermometers.
By Arthur A. Eambaut, M,A., D.Sc, Radcliffe Observer.
Communicated by E. H. Griffiths, F.RS.
Received May 17,— Read June 21, 1900.
[Plates 1, 2.]
Description of the Apparatus and Mode of Reduction of the Observations of
Earth Temperatures.
The instruments with which the earthtemperatures given in this paper were
observed, were five platinumresistance thermometers of the Callendar and Griffiths
pattern,* made by the Cambridge Scientific Instrument Company. These were
purchased by the late Mr. Stone, and were placed in position under his direction
shortly before his death.
The method of platinum thermometry seemed to be particularly suitable for this
class of work, on account of the immunity it enjoys from certain errors attending
the use of the longstemmed mercurial or spirit thermometers ordinarily employed
for underground temperatures.
A higher degree of accuracy might, therefore, reasonably be expected, and the
discussion which follows of the first complete year's observations at the Radcliffe
Observatory shows, I think, that this anticipation has been justified. Some
discrepancies between theory and observation no doubt appear, but they are of a
character which seems to indicate a difference between the assumptions on which
the theory is based and the conditions actually prevailing in the stratum of gravel
in which the thermometers are buried, rather than thermometric errors affecting
the observations themselves.
The thermometers are inserted in undisturbed gravel, the first four lying one
under the other, in a vertical plane beneath the grass of the south lawn, and within
a few feet of the Stevenson screen in which the dry and wet bulb, and the maximum
and minimum, thermometers are suspended.
In order that the thermometers might lie in practically unbroken ground, the
following method of placing them was adopted. A pit was dug at the edge of the
* See the Cambridge Scientific Instrument Ck)mpany's " Descriptive List of Instruments,^' page 20.
2 H 2 22.11.1900
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236 DE. A. A. RAMBAUT ON XJNDERGROUND TEMPERATURE AT OXFORD
grass about 5 feet long by 4 feet wide. One edge of the pit coincided with the
edge of the grass plot, and the corresponding side of the pit was made as nearly
vertical as possible. Into this vertical face four iron tubes were driven horizontally,
the tubes being formed with spikes at their ends to facilitate this operation. The
tubes are 4 feet long, and into them the thermometers were inserted with the leads
attached, the mouths of the tubes were sealed up with tow and red lead, and the pit
filled in.
The first four thermometers were placed at depths of (approximately) 6 inches,
1 foot G inches, 3 feet 6 inches, and 6 feet respectively ; but Mr. Stone soon saw the
advisability of placing another at a lower level, and intended to have gone to a
depth of 20 feet. But as water was met with at a depth of 10 feet 6 inches, he
decided to place it just above the water level, at a depth of 10 feet.
This thermometer was buried, not directly under the four earlier ones, but in a
separate pit at the other side of the Stevenson screen. This was apparently done
to avoid disturbing the leads of the thermometers which were already in position,
but it would have been rather more satisfactory if all had been placed in the same
vertical plane.
It is also, perhaps, to be regretted that one or two similar thermometers were not
buried to considerably greater depths. The presence of water, however, complicated
matters and introduced conditions different from those which prevailed in the dry
gravel above. It is not, for example, to be supposed that the thermal conductivity
or the difiiisivity of permanently waterlogged gravel would be the same as that of
the drier material above it. Hence it would appear necessary to put at least two
thermometers below the permanent waterlevel in order to study the flow of heat
under such circumstances. Besides, it is highly probable that the gravel stratum
is not very much thicker than. 10 feet. Excavations in the neighbourhood show that
the blue Oxford clay is likely to be met with at any depth below 12 feet from the
surface, and in this, of course, the thermal conditions would be likely to prove wholly
different from those in the gravel.
The actual depths of the various thermometers as measured in October, 1898
(when the pits were standing open to enable us to restandardise the thermometers)
were as follows : —
Thermometer .12 3 4 5
Depth ... 6^ in. 1 ft. 6 in. 3 ft. 6^ in. 5 ft. 8^ in. 9 ft. 1 1 in.
These thermometei's, with the Callendai* and GriflEiths resistance box, which
could be connected with each thermometer through a switchboard, had been set up
as I have stated, shortly before Mr. Stone's death.
On my appointment to the post of Radcliffe Observer, I took an early opportunity
of examining the apparatus, and partly with a view of familiarising myself with all
its details, I proceeded to determine the comparative values of the coils, and to
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AS DETERMINED BY FIVE PLATINUMRESISTANCE THERMOMETERS. 237
restandardise a spare thermometer which was kept in the observing room for general
purposes.
This examination led to the discovery of discrepancies in the readings of the
apparatus which troubled me for a long time, and which necessitated a large number
of experiments extending at intervals over the greater part of a year before they
were traced to their sources and eliminated.
In this part of the work I have to acknowledge the very generous help and advice
of Mr. E. H. Griffiths, F.RS., who was kind enough to come to Oxford on more
than one occasion to place his experience at our disposal, and who, at one stage of
the investigation, took the resistance box and spare thermometer to Cambridge to
subject them to a prolonged examination in his own laboratory.
These discrepancies, though serious in view of the accuracy which we had reason
to expect from the apparatus, were still small quantities confined within one or two
tenths of a centigrade degree. They were, for the most part, traced eventually to
uncertainties in the contacts at the switchboard, and a want of perfect insulation in
the older leads. These consisted of four indiarubber covered wires which, in the
underground portion, passed through leaden pipes, but within the observing room
were without the leaden covering. It was found that these were very susceptible to
damp, and that the insulation fell away very rapidly when there was much moisture
in the air, thus giving rise to very puzzling and troublesome discrepancies.
In September, 1898, the switchboard was improved and new composition cable
leads substituted, which extended without interruption from the thermometers right
up to the switchboard. Since these changes were eflfected the discrepancies have
ceased to appear, except on one occasion (viz., October 27, 1899), when it was found
that the short flexible lead from the switchboard to the resistance box was
thoroughly damp. On lighting a fire in the observing room to dry the covering of
this lead, the irregularities disappeared.
Since that date up till the end of March of this year (1900) I have kept a gas
light burning continuously in the room, to prevent the deposition of moisture, and
have experienced no ftirther trouble of the sort.
The resistance box is in its general design similar to that described by
Mr. Griffiths,* but simplified to suit the particular class of work for which it
was intended.
It is provided with three principal coils A, B, C, whose nominal values are
20, 40 and 80 box units respectively, a box unit being about 01 ohm. There are
two additional coils, one for the calibration of the bridge wire, and another, which
we have called the " concealed coil," whose value is about 240 box units, which was
inserted for convenience to balance approximately the resistance of the thermometers
at 0° C. when the coil A was also in the circuit, so that the reading of the bridge
wire under these circumstances might be as nearly zero as possible.
* ' Nataroi* Noyember 14, 1895.
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BY FIVE PLATINUMRESISTANCE THERMOMETERS. 239
oscope is placed on a window ledge to the right of the
position that the observer can manipulate the commutator
1 of the current without removing his eye from the eye
!it of the apparatus is shown in fig. 1. To the right is
oroscope ; underneath in front is the commutator, and
. On the extreme left is the switchboard, and in the
a small electric motor for stirring the oil in which the
d.
the apparatus the method described by Mr. Griffiths in
1895, was in the main followed. The temperature coeffi
\ Griffiths when the apparatus was under examination
Cambridge. Two separate determinations made in 1898
e the following results : —
Range of Temp. Temp. Coeff.
27 . . 9°18 0000242
} 8 . . 12 51 0000240
bservations, the value 0*00024 has been adopted. The
been borne out by subsequent observations in several
ample, the invariable steadiness in the changes of No. 5,
erature of the box, indicated a high degree of accuracy
instant.
he coil values and the unit of the bridge wire scale, the
lade at the Radcliffe Observatory : —
)51
BA= 19851
A = 19603
)46
19849
•600
143
19848
•601
—
19853
602

19847
47 19850 19601
Mr. Griffiths* paper referred to above,
= 80158
= 39 '979 >mean box units.
= 19863 I
•idge ^wire is equal to
I '0134 mean box units.
ible giving the correction for the particular arrange
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AS ICmLJUMT
In iLis eTT«ressJ c K» :s
the i^ts:st.ii:>r of :be " vi:
that orII in tbe i^:L.i 1
mined fi>:»cn tiie o'teerrh::
OMnbined with the cccst.^:.
sponding resistances R a:.
Thus, if X be the ralue
wire when the themior/.r:
Iwidge wire when it is in.^
the total resistances in t:.
ratio of these resist^LCc^
used in the oonstructioa
03872
The values of X found
and 6, 1898, for the purpc*
Thermci
1
4
5
A
For any arrangement of
the total resistance in the
correction for temperature
We thus find the folic
been used in the observat
VOL. cxcv. — A,
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rtVE PLATINUM^RESISTANCE THERMOMETERS. 241
^efficient, 0*00024, as determined by Mr. Griffiths,
.^ = R, X 000024 X (^  14°).
total resistance in the circuit, and since this includes
id coil," we require to know approximately the value of
3 of the equation. This is, perhaps, most easily deter
)f the thermometers themselves at 100° C. and 0° C,
blue found by Mr. Griffiths for the ratio of the corre
his coil, Tq that of the other coils in use and the bridge
s packed in melting ice, and r^ that of the coils and
3d in steam, reduced to mean box units at 14° C, then
.vo cases are, X + ^'i and X + ^o> and if we take the
be 1*3872,* as found by Mr. Griffiths for the wire
X + r
this instrument, then rr— —  = 1'3872, and therefore
^ + ^0
this way from the observations made on October 4, 5,
)f standardising the thermometers, are as follows : —
jr. X.
24065
65
'77
77
60
65
Mean 24068
lis (Y) and any bridge wire reading (R) we have therefore
cuit, X + Y + R> and the coefficient of {d — 14°) in the
(X + Y + R) X 000024.
ng table for the two different arrangements which liave
s: —
'Nature,' November 14, 1895, p. 45.
2 I
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AS DETERMINED Bl
Thermometer .
1.
1898, October 4
5
101286
Adopted values ,
10129
To those who have standa
the separate results for the
however, he pointed out t
while they were sealed up
attached ; hence it was imj
along these tuhes, but for re
take further precaution agai
mining the zero points the
any error arising from this c
One of the most import?
degree of permanence in i
intervals of time. It was tl
found in my first observe
previously at the time that
all the thermometers exhni
apparatus. This examine ti
insulation of which was fou
Another series of discrej
the switchboard. In the (
lead firom the resistance Ik
also were led the four bn
By having the steel pron
finnly pressed against the 1
was experienced ; and, sine
the same cause. It has b
time with the four steel i
check on the character of t
Taking advantage of a
thermometer No. 1 (6 inch
a year 8 continuous obser
year, the zero point of thi
less than 0°005 C, the ac
I]
andii
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FIVE PLATINUMRESISTANCE THERMOMETEES. 243
Temperature of Steam.
2.
3.
4.
6,
101474
101610
589
101389
367
101179
10147
10160
10138
10118
Used naked platinum thermometers the discrepancies in
3. p/s of Nos. 3 and 4 may appear larga It should,
it it was necessary to standardise these instruments
n strong brass tubes with heavy leadencovered leads
•ssible to eliminate altogether the effects of conduction
sons given on p. 244 it was not considered necessary to
st the small errors arising from this cause. In deter
thermometers were placed in a trough 3 feet long, and
,use was very much diminished.
it considerations in connection with this subject is the
le fundamental points as determined at considerable
3 occurrence of discrepancies between the values which I
dons and those determined about a year and a half
he instruments were set up, which induced me to have
3d and to make a thorough reexamination of the whole
n led eventually to my discarding the original leads, the
d to fall off very much when they became damp,
mcies was traced to an uncertainty in the contacts at
[•iginal form the four steel prongs in which the fourfold
c terminates, were inserted into mercury cups into which
58 strips to which the thermometer leads were soldered.
s amalgamated, and adding springs to keep each prong
rass strips immersed in the mercury, a great improvement
I this change was effected, we have had no trouble from
3n the habit, too, to make the observations from time to
ongs in both positions, which affords a very satisfactory
e contacts.
visit from Mr. Griffiths on October 6, 1899, I had
s) dug up, and we examined its zero point after exactly
ations. Determined in the same way as in the previous
thermometer was found to agree with the earlier value to
lal values being
1898 0'806
1899 0'802
2 I 2
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AS DETERMINEE
A complete determina
of the sunken thermomet
(1) The balancing of
and the temper;
(2) To R is to be adde
from Table I.
(3) The correction to
Table XL
(4) The reduction to
Table III. mult
(5) The correction froi
It only remains to re
the air scale.
The relation connectin
in which pt is the platii
a constant.
For a completely inde
the resistance at some
but the experiments of '
value of 8 varies from on
particular sample of wii
are given in Mr. Griffit
The value of 8 for th
mined at Cambridge t(
apparatus for the deter
doubtless have been adv
this constant. Since, Ik
variations of earth tern
within that range the cci
0*050 (which is quite m
Writing ^« + dforti
0'003, its square may be
T being written £cvptil(
* 'PhiL Trans./ A, 18S7.
t Cy. the Report of the C
Practical Standards for use in
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' FIVE PLATINUMEESISTANCE THEEMOMETEES. 245
of temperature on the platiuum scale by means of one
is, therefore, reduced to the following simple steps : —
galvanometer and reading of the bridge wire scale (R)
L'e {0) of the box.
le correction for the particular arrangement of coils used,
luce the bridge wire reading to mean box units, from
indard temperature (14°). The quantity taken from
ed by {0 —14) gives this correction,
^able IV.
3e the temperature thus expressed from the platinum to
bese two, established by Professor Callendar,* is
'p"ii^)'U w
1 temperature, t the temperature on the air scale, and 8
ident standardisation it would be necessary to determine
rd known temperature in order to obtain the value of 8,
LLENDAR and Griffiths have shown that although the
ipecimen of platinum to another, it is a constant for any
References to the original papers bearing on this point
* article in ' Nature ' cited above.
)articular wire used in the Oxford instrument was deter
)e l'512.t If it were intended to employ the Oxford
Qation of temperatures over a very wide range, it would
ble to make an independent determination of the value of
3ver, the range —15° C. to +25° C. will cover all the
matures with which alone we are here concerned, and since
iction does not amount to as much as 0'3, an error of even
lissible) in the value of 8 would not affect our results,
equation (6), and remarking that since dj 100 is less than
jglected, we find
= 8(t^  t)/{1 + (1  2r)8/l00}
hiittee of the British Association for improving the Construction of
metrical Measurements. Bradford, 1900. [September 16, 1900.]
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X^ iS
Ulkf
^1^ rr^ fcrc :z. lijt .
results. ATii Jis tie : f^r
disdiict adv^LHt^o? Ir. rv:
On aocoant of the ir.'i
carded them alt* gether,
pcfftioDS which are alte: :
vations are taken only oi
of days in each divisir.
lengths as small as {k^^
thirty and thirtyone c
intercalating the extra c
♦ Professor W. Thomsox,
Eoy. Soc. Edin./ voL 22, p.
t ' Greenwich Observatior
t Professor Everett, ' T:
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LMIMD BY FIVE PLATINUMRESISTANCE THERMOMETERS. 247
. . 9550."
Sept.
22 . .
1442 C.
. . 959
. . 963
•CoaA.
23 .
24 .
. 1441
1441
. . 966
25 .
1440
. . 970
26 .
. 1439
. . 975
. . 980
•CoaR
27 .
28 .
. 1437
. 1435
, . 982 .
29 .
. 1434
SCoilR
I
Coil A.
Discussion of the Obso^^atiojis.
the discussion of the observations is to group them into monthly
) to deduce the harmonic expressions which will represent the
ermometer throughout the year.*
be work I have adopted the Fahrenheit scale, as the observations
educed to this scale for comparison with our other meteorological
observations of the same kind at Greenwicht and Edinburgh
sor Everett are expressed in the same scale, there seemed to be a
in retaining it,
e inequality in the lengths of the calendar months I have dis
bher, and, as far as possible, have divided the year into twelve
iltemately thirty and thirtyone days in length. As the obser
ily once a day, it is of course necessary to have an integer number
vrision, but the following scheme makes the differences in their
I possible, and with one exception, that of January, alternately
►ne days. In Leap Year this exception would be removed by
bra day in January, instead of February.
3N, " On the Reduction of Observations of Underground Temperature," * Trans.
p. 409.
^ions/ 1860 (cxciii.).
Trans. Roy. Soc. Edin.,' vol. 22, p, 429.
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AS DETERMINE]
McClellan in the or
Mr. WiCKHAM or Mr.
and experience, and, as
able degree of precision.
These means were d
any modification or cor
to above, except on one
flexible lead was found
room.
This was indicated b
10 feet thermometer, \
steadily, that its readin
onetwentieth of a degr
drying the lead, howeve
the subsequent readings
as before the diBcrepancj
As the dampness of
and in no way affected
take an interpolated vali
the difference between
added as a correction to j
This particular case i]
deep sunk thermometer
apparatua
The monthly means an
periods of two months ea
been selected, as the fir
the 10 feet thermometer
in the indications of this
amplitude of a wave is
thermometer to the one 1
In fig. 3 are given
surface, deduced from th(
The harmonic expres
thermometer throughout
e
or 0^ao+t
where t denotes the time
VOL. cxcv. — A.
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MINED BY FIVE PLATINUMRESISTANCE THERMOMETEES. 249
he ordinary routine, but during his vacation, or on Sundays,
• Mr. Robinson took his place. All three are observers of skill
id, as the results seem to show, the observations are of a remark
iision.
ere deduced from the observations as directly obtained without
)r correction, other than those taken from the tables referred
1 one date— October 27 — when, as I have mentioned, the short
mnd to be affected by the dampness of the air in the observing
led by a sudden change of about 0°*13 R in the reading of the
er, which, under] ordinary circumstances changes so slowly and
3ading on any day might be predicted with certainty to within
degree from the readings of two or three days preceding. On
^ever, the abnormal readings disappeared by the next day, and
ings of this thermometer were foimd to lie along the same curve
•ancy had arisen.
of the lead disturbed only the reading of the resistance box
cted the thermometers themselves, we were, therefore, able to
value for the reading of No. 5 as a standard of comparison, and
)en this and the actuallyobserved readings, viz. : 0°*13, was
to all observations made on that day.
je illustrates very well the protection which the readings of a
ter afford against sudden changes occurring unobserved in the
1 are graphically represented in fig. 2 ; the daily readings for two
3 each are exhibited in Plates 1 and 2. These two periods have
first includes the minimum and the second the maximum of
er, and both illustrate very well the steadiness of the changes
his instrument, and exhibit also the manner in which both the
is dinainished, and its phase retarded in passing from one
e below it.
n the mean monthly temperature gradients beneath the
:he same figures.
ression to represent the temperature of any particular
it the year will be
^ = aQ + a^ cos \t 4" % ^^ 2X« + &c.
+ hi sin \t + 62 sin 2X« + &c (c)
Pi sin (Xe + El) + P2 sin (2Xt + E^) + &c (d)
e represented as the fraction of a year, and X is equal to 2ir.
2 K
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AS DETEEMIKEI
i
o
08 4a
u
o ^5
»4
I
1 ^'^
?
'J
H
1
1
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•LATINUMRESISTANCE THERMOMETEES.
251
K) X». «J « K •O .«!> 5f'
•399^ Ul tf9c/9Q
2k 2
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DETERMINED
are here dealin
nuch importanc
juracy with ^
the formulae i
between then
iriBon of Compi
Thermometer.
January . . .
February
March.
April .
May .
June .
July .
August
September .
October .
November . .
December . .
; appears that w
r the formulae, 1
^ially in the ca^
coselves felt to a
rfece of the gn
are sunk, beinj
a fairly uniform
at will be repres
K denotes the
ter below the bi
ion of this is
'ing this expres
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:>LATINUM^RESISTAITCE THERMOMETEKS.
253
observations of a single year, it would be unsafe
3 deducible from the smaller terms.
Dbservations are represented by the first three
by the following table, which contains the
ily temperatures as computed and those actually
bserved Mean Monthly Temperatures, C — O.
2
3
4
■ ■■
5
0°86
 d'55
 0°32
0°00
018
+ 007
+ 015
+ 005
+ 071
+ 048
+ 022
+ 002
070
048
019
000
+ 091
+ 025
+ 001
005
102
062
024
002
+ 115
+ 091
+ 049
+ 006
110
044
007
+ 010
015
069
065
023
+ 187
+ 129
+ 073
+ 007
214
107
031
+ 018 .
+ 162
+ 075
+ 018
018
iree deeper thermometers are fairly well repre
lonsiderable differences in the two upper ones.
L, is largely due to the diiUTial variations which
bout 3 feet.
le neighbourhood of the spot where the ther
lately level and the gravel being, as far as we
for a considerable distance in all directions, the
ourier's equation
0/dx^ = d0/dt . . (c)
of the gravel, and x denotes the depth of a
and 2af,finK = — nX.
Vt = ^'
le series {d) given oa page 249 we have
' and E, = /8,a! + y.
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aiMINE]
Values c
Them
com
No. 5 a
„ 6
„ B
„ i
» 4
„ 3
Mear
d the re
y short
uried ir
contaii
• ^/1^/K
I those
dividua
ote hov
from tl
possibly
h separ
). 5 bei
the oth
"when
gravel"
n next
f close
y^early
duced
)recisic
ally la
dedu
may p
' whic]
of th
it at
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PLATINUMRESISTANCE THERMOMETERS.
255
iuced from the Halfyearly Wave.
From diminution
of amplitude.
From retardation
of phase.
0U45
•1393
•1306
•1287
•1164
•1033
01236
•1187
•1129
•1092
•1022
•0949
01271
01102
of both , . . 01187
ed from the readings of No. 1, as they seem too
iations to aflford reliable results. This thermo
soil which is of quite a different character from
ler thermometers.
rom the annual wave are, of course, much more
rom the halfyearly wave, and the larger discrep
om the latter are not surprising. It is, however
roborate the others, showing, for instance, larger
son of Nos. 5 and 4 than from that of any other
a smaller value of k for the stratum of gravel about
two thermometers, than for the higher strata, or
it some distance (9 feet 6 inches) from the vertical
were open, no very critical examination of the
nt depths was made ; but it is proposed to repair
)meters are dug up.
of the mean values of x/v/k derived from the
ry remarkable (especially in view of the fact that
servations of a single year), and seems to indicate
servations.
found from the diminution of amplitude, as com
he retardation of phase must be traced to some
ue to the proximity of the Observatory building,
1 at a distance of 36 feet from the thermometers.
)eneath the buildings would in all probability be
epth beneath the exposed surface. There would,
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5
4
3
2
values i
a givei
to any ^
main unc
oreticalJ^
i^e may c
acted. T
to 0°01 1
)th we h;
1 therefore
as the de
•^1 =
or the half
X. =
at which
0^1 F. are
deals with
[)aucles l)et
lave been
ancles are
r a periodic
to he dimi
physical ai
ource of in
lely, thernK
:he liquid
—A,
Digitized by VnOOQ iC
PLATINUMRESISTANCE THEKMOMETERS. 257
^e, we find
1
9
7
2 = amplitude of halfyearly wave at surface.
Df equations (/) we can determine the value of r,
P,„ or the depth at which the amplitude of the
, on the hypothesis that the conditions prevailing
greater depths.
10 invariable layer so long as equations (/) are
it an annual variation of 0^'02 F. is less than can
, therefore, at which the amplitude of the annual
ntents and purposes be considered as invariable.
P^ = — 2 (M being the modulus of common
+ log PoO/M^Wk.
3h the amplitude of the annual wave is reduced
rich feet = 66*0 English feet,
e,
ich feet = 360 English feet.
al and halfyearly waves are reduced to an
I similar way to be 45*3 and 21*4 English feet
tions of a single year, and the results accordingly
J and observations which, although they are less
pected, are greater than one would like to see.
to the fact that the temperature variations are
as the theory supposes, and as such they might
he mean of a number of years, and partly to
I the surface.
►nsi<lered by Lord Kelvin in his paper, referred
►rs arising from the uncertainty as to the tem
; stems of the thermometers used in Professor
*> T
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\
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a d
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Phil. Trans., A, vol. 1.95, PL i.
It 17 16 19 SO a^
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Phil. Trans,, A, vol. 1.95, PL i.
/d 17 18 i9 20 Sil 22 as B^ £5 25 27 as B9 30
9 /7 /a /9 20 2/ 22 23 2^ 25 26 27 26 293C
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VIL The Diffusion of lous ^>
stance J i
By John S. Townsend, M
Fellow of
Conimunicato
Rect
A GENERAL method of findiii
described in a previous paper,^
with ions produced by Rontge
with ions produced by a radi(
violet light. Tlie principle of
from observations on the los
tubing.
The experiments were arran
greater than the loss due to t>
effects which must be consider<
binatlon which occurs when tli
gas ; and the effect due to the
most of the ions are chargt'f
necessary either to correct for
the experiments so that the lev
The present paper is divid
investigation of the relative ii
and mutual repulsion in causii
and the results of the experinn
by ultraviolet light, and by j)
respectively. The conclusion^
Section V.
In the previous paper we h\
distributed throughout a gas,
* JuHN S. Tow
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[ 259 ]
id in Air by the Action of a Radioactive Sub
violet Light and Point Discharges.
hrk'Maxwell Student, Cavendish Lahordiory,
/ College, Caml/iidge.
ro/essor J. J. Thomson, F.Ii.S.
J 17,— Rend June 14, 1900. .
rate of diffusion of ions into a gas has been
1 account was there given of the results obtained
The present paper gives the results obtained
substance, by point discharges, and by ultra
tliod consists in calculating the rate of diffusion
nductivity of a gas as it passes along metal
}hat the loss due to diffusion should be much
uses. In order to ensure this, there are two
iiig the dimensions of the tubing : the recom
x)th positive and negative ions present in the
repulsion of the ions which takes place when
electricity of the same sign. It is therefore
Lirces of error or to arrange the conditions of
ductivity due to these causes is negligible,
five sections. The first section contains an
e of the processes of diffusion, recombination,
conductivity. The descriptions of apparatus,
B on ions produced by a radioactive substance,
larges, are given in Sections II., III., and IV.
l'a^vn from the experiments are discussed in
CTION I.
that when a number of ions. A, are uniformly
is entering metal tubing, the ratio R, of the
\. Trans,' A, vol. 193, 1899, p. 129.
L 2 8.12.1900
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BY THE ACTtn
positive and negative ions ai»
experiments on diffusion, we a
very closely those produced n;
therefore assume that the ia\\ •
in the two causes. Tlie method
explained in the previous pa{M
the loss due to reconibinatH»n
sides.
The time, Zi/V, in the ex\)t
second, the radius of the tuhi
finer tubing (a = 5 millini.).
reduced to ^. The numl)er,
N to N/9.
The radioactive sulistance
the radiation proceeding fron
smallest that was used in t
therefore assume that in tht»
not affect the value of y to th
When a gas contains ions i
of ultraviolet light on a nieti
arising from the electric densi
It would be difficult to find tl
of ions in a tube while dift\isi(
to the error it introduces.
Let us consider the c<\Ke ol*
owing to the motion of the io
we suppose that no diffusion
electrification at any jx)iiit is
Pq being the initial density, s
by unit electrostatic force,
Po t<> P
The proportion of ions lost,
t John
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A lUDlOACTIVE SUBSTA^^Cfi, ETC. 2()1
imultaneously in the gas. From the results of the
to conclude that the ions thus produced resemble
tgen rays, and carry the same charge. We will
rning the recombination will not be much different
nding the correction for recombination has been
[t was there shown that for small conductivities
bout 4 per cent, of the loss due to diffusion to the
;s made with Rontgen rays was about Ybth of a
ig 1 5 millims. A new apparatus was made with
without altering KZ^/a^V, the value of Zj/V is
ons which recombine is similarly reduced from
tained in a sealed glass tube, which cut down
s to produce densities of ionisation less than the
jriments made with Rontgen rays. We may
experiments the process of recombination does
of '5 per cent.
ual Repulsion.
n (as in the case of ions produced by the action
)r by a point discharge), the electrostatic field
letimes suflBcient to exert a considerable force,
imount that this effect contributes to the loss
ng place, but it is easy to find an upper limit
I gas in a metal tube losing its electrification
iiies of force from the axis to the surface. If
)lace, it is easy to showt that the density of
:he formula —
Po
1 f 4c'Trupf/ '
Liforiii, u the velocity of an ion when acted on
time during which the density falls from
practically AnpoUt when the loss is small.
>, ' Phil. Mag.,* Juiie, 1898.
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Dounda
r down
d by tl
a^ to tl
th air at
he mova
be of tl
3 tightly
leading t(
atus. W
3 tubes w(
•elocity of
gasometei
3 suljstanc
r), and th<
of thin gl
f wire 8up]
it by the ac
through th(
C was seale
i radioactiv<
ebonite supj
haken when
was eonnecte
eing to earth.
the other pa
rth. The rod
metal screens
le electrometer
abe A is in n
ipt E, there is ii
es T into the s]);
dctrode, and a d
ordinary conditi*
turbulent moti<
itial difference (
periment never <
ionisation and \
larging A to 40
one sign are en
:o the number of
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F A RADIOACTIVE SUBSTANCE, ETC. 263
the disc. When either set of tubes was pushed
arge tube divides itself equally among the twenty
bes T. The disc a^ was soldered to the front of the
nt of the tube B^j.
ospheric pressure the stream of air was obtained by
ylinder of a gasometer. For experiments with dry
asometer was connected to wide tubes of calcium
ked with glasswool was put between the drying
so that particles of dust should not be carried into
it was desired to make experiments with moist air,
emoved, and long tubes half filled with water were
air along the tubes T could be varied by changing
as obtained from E. de Haen (Chemische Fabrik,
eparation labelled " Radioactive Substance A " was
containing some of the radioactive substance, was
s inside the tube A as shown in the figure. The
substance was much more intense than uranium rays,
iss tube was strong enough to ionize the surrounding
1 order to prevent any moisture from coming into
ibstance, which was deliquescent. The tube A was
5, S, to the top of a heavy box, so that the tube C
tubes B and IB^ ^^® fixed in position,
o one terminal of a battery of forty lead cells, the
i'he rod F was connected to one pair of quadrants of
f quadrants and the case of the electrometer being
and the wire connecting it to the electrometer were
\ that external electric charges should not give any
tie.
illic connection with all the parts of the diflRision
electric force acting on a stream of gas until it comes
between E and B. The air takes about one second
3rence of potential of a few volts between E and B
J, suffice to collect all the ions of one sign on E, but
of the gas as it escapes from the tubes T, a much
) volts) was used. The potential of the electrode
needed 1 or 2 volts. It was found under similar
ocity of air that the electrometer deflection was not
^Its instead of 80. We therefore conclude that all
^cted on E, so that the electrometer deflection is
)ns that come through the tubes T,
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BY THE ACTiC
nds of the tubes 1
tube A, and was
A CQuld be found,
apillary tubes K,
le admission of ai
ary tubing was coi
made with wide t
vessels W^ and W.
er to obtain a streai
between suitable lin]
iusted until the pn
^as connected to tht
IS adjusted so as to
a open to the air. 1
as turned on for a
3re desirable necessil
out by the waterpu
f rose at the rate of a
the pressure was as
jnt The velocity V
>f air that escaped fr
5X0V,— A.
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N OP A feAbiOACtlVE SlJiBSTANCE, ETC.
265
I and Bjj. A short brass tube L was soldered near the
onnected to the manometer M, so that the pressure of
The air from the room was admitted to the apparatus
laving first passed through a tube of glaisswool G, to
Y dust which might alter the resistance of the tubing,
lected to the drying tubes, and the rest of the connec
:bing. The tube u leading from B was connected to
which were exhausted by means of a waterpump.
^£
^i>^
1
~ g y
Fig E.
of air through A at a given pressure P, and with a
s, the stopcock Si was closed and the whole apparatus
mre was a few millims. below the pressure P. The
elivery tube of a gasometer, the movable cylinder of
3 on the point of moving downwards when the gaso
j vessel Wg was then connected with the waterpump,
V minutes. The velocities V (through the tubing T)
ed a larger supply through the apparatus than could
) so that the pressure, as shown by the manometer,
ut 3 milliius. per minute. The stopcock Sj was turned
ich above P as it was below P at the beginning of the
the tubes T can be accurately found by observing the
the gasometer and the time during which the stop
2 M
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" THE ACT
roa of men
a the tempe
jitive Ions ii
V 1 J
344 Vt
387 55
420 4C
410 30
682 20
itive Ions in
V. P.
1
368
77:
430
40(
609 20(
n bet^
icaUy I
^een t
»y me?
1
^
X
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N OF A RADIOACTIVE SUBSTANCE, ETC.
267
ry ; V is the mean velocity in the tubing Tj in centime,
ture of the air during the experiment : —
Dry Air.
Table II. — Negative Ions in Dry Air.
a
»i.
na.
V.
P.
e.
19
634
1386
344
772
19
13
430
938
387
650
13
16
248
680
420
400
16
13
106
399
410
300
13
1 12
76
315 582
200
12
[cist Air. Table IV. — Negative Ions in Moist Air.
a
18
11
95
«1.
«2.
V.
P.
e.
711
135
368
772
18
210
663
430
440
11
76
271
609
200
96
I ratio y {= ^'i/wg) and the coefficient of diffusion can
3 of a curve representing equation 2, Section I. The
9 i'O hi i'B i'3 h^ h^
3 M 3
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BY THE ACT]
•es, Si and S^ were
A quartzplate,
e the joint airtij
Jaced in the wii
^
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ON OF A RADIOACITVE SUBSTANCE, ETC.
269
saddled on to it, each of them surrounding one of the
J, was fixed to the end of S^ by means of sealing wax,
ht, and a piece of wire gauze, having the same curvature
dow Wi, completely filling it. A piece of zinc, Z, of the
3ce of brass which was cut out of the window Wg, was fixed
sed through the ebonite disc D. The disc fitted tightly
joint was made airtight. The zinc did not touch the
il relative to A could be varied as desired. When ultra
he quartz and the gauze, it falls on the zinc, and negative
face of the metal. Some of these ions can be sent into a
^ A by lowering the potential of the zinc relative to A.
battery, H, was insulated and its positive terminal was
.tive terminal to R.
minium wires was used as the source of ultraviolet light,
g the spark was contained inside a box covered with lead,
through which the light from the spark fell on the quartz
nals of the secondary of a Ruhmkorfi* coil was connected
joyden jar, and the other terminal to the inner coating,
e coil, and the discharge took place across the sparkgap
ium wires. The air in the neighbourhood of the spark
id, so that it was found necessary to pack wool round the
rifled air from coming into the neighbourhood of the rod F
: to the electrometer. When this precaution was taken it
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BY THE ACT
The density of ionizatioi
has to be polished from ti
smalL In the above exp
the third experiment with
gas in the tubes T^ is 5^7
apparatus per minute was
On standardising the el
sponded to a charge of Oi
fication p was therefore {
Section L that the product
due to selfrepulsion shouL
the sides. In the present
need be made for the loss c
Secttion I\
In order to make the a
charge, the changes shown
diameter) were made in th<
F%g.4
^h
'h
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OP A RADIOACTIVE SUBSTANCE, ETC.
271
pends greatly on the state of the zinc surface, which
bo time in order that the ionization should not be too
ents the greatest density of electrification occurs in
gas. The mean time, ^, spent by any portion of the
ad. The total volume of gas that passed through the
> c.c.
ometer, it was found that each scale division corre
electrostatic unit. The mean density of the electri
< 10"^ electrostatic unit per C.C. We have shown in
C t must be less than 10"* in order that the loss of ions
B less than 1 per cent, of the loss due to diffusion to
e the product pX* is '9 X 10"^\ so that no correction
to selfrepulsion.
Ions produced by the Point Discharge.
stratus suitable for experimenting with the point dis
L fig. 4 were made. Two circular holes (1*6 centims. in
ibe A, and two tubes, Q and B, of the same diameter as
;,£^
Fig. 6.
w
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\r *  •
S.^:^ as 1
J>anie as
iitinis, fn»i
xjx^riineiit
burth ex J I
:es place i
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S OF A tVPKV.ACriYK srRSTANCR KTX\ l^7S
I stcvl \\i\\t in the tiiK* Q, the jxMiit Knr*g at :V»e
e ui:der5«ine ivndit unis as exj>oriment K exivpt tV,ai
• :\iVr that the ::[5u^ v^houM have a siiialler eUvtnt;vxu:ov,
vriinents 2 and 3, with a platimim jvnnt sul^titiuoil
.< held in the tul^ A hi the [xv^tiou shown ux ti^j. o,
ermient 1, except that the }x>int n>5U5 dniwu up tho
111 the aperture iu A*
»iily ones iu which the efteot itf st*lfn*pxilslon may
;s of ions in the tube Tj, so that the n^Uuos of K
lav be a little too big.
*ent. which occurs lietweeu the vahuvs obtainiHt in
»s obtained iu experiments 2, 3, and 4, is pix>lvibly
ifference in the ions,
iiu of air in A from a x>iut some distance up tho
5, as Experiment 6 sho\>T5 that they dittuse nunv
Xegati
ive Ions
in Dry
Air.
«i.
fi^.
V.
K.
■2
138
337
•0382
62
165
326
•0367
<5 I
150 1
323 ' 0368
■2 i
1
I
160
342
•0324
[)oiiit ill the tul)e Q, tho point Wnug at tho
1, with a platinum point Hul>Hlitutod for [\\v
I the tube A, as shown in fi^. 5.
7, exct»pt that the j)oint was drawn uj) \\\\'
iie ill A.
cjilly the Hamo vahioH for tho ooollicioni of
vH that larger ions aio imxhuvd when llio
tube Q.
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BY THE AC
The theory of the inter
iversely proportional to tl
een confirmed by the exjv
Section IL show that the 1
onsists of ions. The prt
ontribute to the total pn
ressnre in this case is the
:e see that between the
iffiision is inversely propo
We conclude firom this t
aries between these linlit^
L
' The experiments on diti'i
ctive substances, and ultn
he same changes arising
oefficients of diffiision of i
ases which are greater thu
The ions produced by th
ther methods, since theii
egative ions in moist air.
Coefficients of Dit
Method.
Rontgen rays
Radioactive siil>stance
Ultraviolet light
Point discharge
Me^
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OF A RADIOACTIVE SUBSTANCE, ETC.
275
IX v..
of Pressure.
an of gases shows that the coefficient of diffusion is
1 pressure of the two diffusing gases. This law has
ts of LoscHMiDT and others.* The results given in
1 be extended to the case where one of the gases
that the ions exert is so small that it does not
\y an amount which could be measured. The total
ire 0^ the gas into which the ions are diffusing, and
as 772 and 200 (millims. of mercury) the rate of
;o the pressure,
size of an ion does not change when the pressure
iced hy Various Methods.
w that the ions produced by Rontgen rays, radio
ight are nearly of the same size, and subject to
) presence of moisture. The following table of
lir shows that there are differences in the various
light arise from experimental errors,
ischarge are larger than those produced by the
diffusion is much slower, except in the case of
ons produced in Air by different Methods.
Dry air.
Moist Air.
) ions.
Negative ions.
Positive ions.
Negative ions.
•035
•043
•032
•043
•036
•041
•043
—
•037
1
•037
•032
•028
•027
•039
•087
Theory of Gases,' Chap. VIII.
? N 2
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[E AC
y that
It of tl
> come
iits of
^ woul<
ficient?
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OF A RADIOACTIVE SUBSTANCE, ETC. 277
sizes of the ions produced by point discharges vary
iparattis in the neighbourhood of the point. What is
definite conclusion with regard to the charges is to
sion and the velocities of ions produced under similar
possible with the apparatus I have used for the deter
[iflEusion, and I hope to be able to make observations
to an accurate determination of Ne.
charge on an ion in a gas in terms of the charge on
is of some importance, as it enables us to obtain
f electricity.
IS a similarity between the minimum subdivisions of
gases.
r the determination of the charge in absolute units
b gases, and since all the determinations depend upon
^reat accuracy cannot be expected. The results show
is of the same order for ions obtained by various
)een obtained by Professor J. J. Thomson for ions
Lnd by ultraviolet light ;t the values are nearly the
^ and 7 X 10"^^ electrostatic unit. These values do
due 5 X 10~^^ which I obtained for the charge on the
en off by electrolysis.;};
;e is the same in all cases, we must assume that the
in order to account for the differences observed in the
LELLAND,§ by examining the velocities of the ions
owing wires, found that the mass attached to the ion
circumstances connected with the ionization. The
es for small differences of temperature of the wire,
llects round an ion is very variable. We would not
dioactive substances would have an effect upon the
icy to collect round a charged ion, but it is possible
Afferent ways by different kinds of rays, so that the
h point discharges in air there are actions taking
e the carrier increase in size. Thus the oxides of
t condense round the charge and lower the rate of
of the ion.
violet light has any effect on dry air, but Wilson
Phil. Mag.,' Dec, 1898.
Phil. Mag.,' Dec, 1899.
* Phil. Mag.,' Feb., 1898.
1), * Camb. Phil. Soc. Proc,' vol. 10, Part VL
* Phil. Trans.,' A, vol 192, 1899.
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279 ]
%re of Metah, (Second Paper.)
Mechanism and Applied Mechanics in the
TER RosENHAiN, B.A., St. John's Colhge,
h Scholar, University of Melbourne.
ead, in Abstract, May 31, 1900.
:s 3—13.]
3resent paper deal principally with the
>ntinuation of the research described in the
' A, voL 193, 1900, pp. 353377). In iron,
•een studied with the aid of the microscope
D, Chakpy, Stead, and RobertsAusten
esult of their labours it is well known that
nent of the crystalline grains of the metal.
1 in tension its crystalline grains become
; when the specimen has been subsequently
all signs of such elongation disappear from
microscope. * In fact it is not generally
Btween the crystalline pattern seen in the
ling. In general, the pattern seen after
ilar specimen before it has been strained,
em produced depend very much on the
trly upon the temperature applied, the time
g. Arnold and Stead have shown that
rge crystals in iron and steel. But even
e is well known to produce complete re
that these changes occur at critical points
the cooling of the metal. These arrest
is natural to suppose that they are evi
of the metal.
! hoped to observe this change taking place
jrimental difficulties of keeping a specimen
J being heated were successfully overcome,
of iron failed.
10.12.1900.
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^E STEUCTUEE OF METALS. 281
r the atmosphere. On leaving the apparatus
ugli the window by means of " vertical " illu
ve itself; as we were content with moderate
n objective of long focus could be used.
, we did not succeed in keeping the polished
» had been reached ; but in the course of our
)n was observed. On beginning an observation,
'' ferrite " grains could be clearly distinguished.
t'^ere then slowly raised by gradually turning on
irst visible change was a dimming of the image,
etely blotted out. This we supposed to be due
3 part of the optical system, but we could not
re further, the image of the crystals reappeared
ting off the reflected light, the metal could be
mating still further, the pattern was rapidly and
5 of the surface ; the metal was now dullred.
ily dark spots appeared, and spread rapidly over
eed at which they spread could, however, be
leating current. The spots appeared well in the
)parent darkening could only be pushed to the
erably higher temperature. On allowing the
isible, either on passing through this range of
i ; nor could the phenomenon be made to recur
d below redness; but, if this was done, the
y in the same specimen. It seems probable that
ance occurs in the metal itself and not merely in
Lis in this film remain entirely unaffected by it,
'ssion that he is looking at an action taking place
arent film. On repeating these observations with
atmospheres than hydrogen, no such phenomenon
to suppose that the phenomenon is a result of
'^drogen and the iron. From its occurrence just
sspond to the arrestpoint, about 487^ C, discovered
^s Research Report,' Inst. Mechan. Engin., 1899).
hydrogen caused the surface of our specimens to
rercome this difficulty by observing the surface of a
ating was again done electrically, either by passing
ecimen, or else by placing the specimen in the centre
and on a piece of terracotta. In both cases the
, the electrodes passing through a sealed cork at the
2 o
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^E STRUCTURE OF METALS. 283
cloned the attempt to observe this process iu
to the study of similar processes of amiealing
iible metals, particularly lead.
ture required to produced recrystallisation in
we observed in specimens of plumbers' sheet
ite nitric acid. When thus treated, ordinary
Laiit crystalline structure on such a large scale
The etched surface shows all the appearances
light on etched crystalline surfaces ; when the
he various crystalline grains in turn, the colour
each grain, but different on different grains.
ich a surface magnified two diameters : these
ring, and must therefore be observed and pho
►lution.
•face reveals a peculiarity in the configuration of
jn to have many remarkably straight boundaries
of parallel boundaries being frequently observed.
ible what we had previously observed in wrought
[uent occurrence of twin crystals. In our earlier
presence had always been readily detected by the
in them by slightly straining the specimen after
observed in sheet lead by this method has been
Phil. Trans.,' A, vol. 193, 1900, Plate 26, fig. 40).
of detecting twins is not available, as the rough
iepth of etching employed make it impossible to
mce of twin lamellsa nevertheless becomes evident
with oblique light. Fig. 3, Plate 3, is a photo
B magnified 40 diameters. The figure illustrates
imination, which has picked out a few isolated
J while neighbouring ones remain almost dark.
luminated grains, a number of dark patches are
 boundaries occurring in parallel sets which are
is instance there are three distinct parallel sets
hey are twin lamellae becomes apparent when the
.ted, thus altering the incidence of the light. As
Qs that were bright become dark, but presently
eviously dark shine out brilliantly, all the bands
ashing out simultaneously. Fig. 4 is a photograph
rotation of about 30°, and illustrateis this appear
'^hich catch the light simultaneously are evidently
which the orientation of the elements has been
other words, they are twin lamellsB.
2 o 2
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CRYSTALLINE STRUCTURE OF METALS. 285
ermine the effect of very severe strain on the crystalline
a soft, ductile metal, plastic deformation may be carried to
le adaptability of the individual crystals to change their
leavage planes may be insuflficient. Careful observation of
f a piece 'of lead under severe compression confirms this
it the crystals are gradually flattened out in proportion to
specimen, but when the " flow " becomes considerable it is
ady very thin and flat, are driven into and through one
ig in a grain or structure which is small, but still entirely
inalogous to what occurs in the fracture of a more brittle
;hat in a more brittle metal, when " slip " has gone so far
rystal, the new siu'faces thereby brought into contact do
cture results ; in lead the freshly exposed surfaces do weld
re, a fact which is associated with the possession, on the
utility. Fig. 6 is a microphotograph showing the crystal
i lead magnified 1 2 diameters, while fig. 7 shows the much
3shly and severelystrained lead magnified 30 diameters,
xperiments with lead, the process of straining was carried
of the metal in a compressiontesting machine, letting
jlock, originally about 1 inch high and f inch diameter
ut I inch thick.
e changes in the crystalline structure of such strained
of taking a series of photographs of a marked area of
;ime during which the metal was exposed either to the
room or vras subjected to special thermal treatment.
iken the surface was thoroughly reetched ; our experi
L had convinced us of the necessity of this proceeding,
pecimens have confirmed the previous experience. In
t in any way produce a visible change in the surface
ad been resorted to, and fairly deep etching is required
tirely. This applies more particularly to the channels
atercrystalline boundaries — ^these may often be seen
yformed pattern, but quite independent of the new
ted of alternate applications of concentrated and very
, where very deep etching was required, an electrolytic
great advantages of dealing with a metal like lead
jrystaA& ; by enabling us to use deep etching it allows
sed vsrith, and it becomes possible to obtain micro
ns, and under oblique light, which exhibit clearly the
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LINE STRUCTURE OF METALS. 287
erefore be taken as no more than an extremely
agnitude with which we are concerned in these
iperatures.
sheetlead showing fairly large crystals as an
perature annealing has continued for a long time,
plying a higher temperature, so as to determine
occur. Our observations show that the metal in
•ately high temperatures, three minutes' exposure
ture of 200° C. being suflficient to produce a great
If the specimen be kept at 200° C. for a long time
it becomes very slow, and ultimately a state is
»erceptible.
inclusive are a series of microphotographs of a
1 4 shows the appearance of a typical specimen of
iange produced by 30 minutes at 200° C. Except
is very diflficult to trace any connection between
ginal. Fig. 16 shows the same surface, reetched
)^ C There has been further change, but not to
n the first halfhour. The change is most marked
side of the figure ; in fig. 1 5 it shows a mottled or
nes filled in in fig. 16, while there is a considerable
ss of the two tonguelike projections that start on
nd.
crystal is seen in fig. 17, which was taken after
^ C. Here another twin band has become evident,
htening of the boundaries has taken place. This
n fig. 18, taken after four days' further annealing.
representing the final state of this specimen, as
1 no further considerable change. This specimen,
ng feature, which we have often observed in other
)r photography occupied the centre of the surface
1 approximately f inch square by ^ inch thick,
marked area did not show by any means the best
1. In this case, as in many others, we found that
ing crystals were formed at or near the edges of
it the same stage as fig. 18, and with the same
irked area, shows the remarkable development of
men.
uch a series of photographs, one consideration must
3nce produced in the appearance of the surface by
f incidence of the light. In spite of the utmost
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CRYSTALLM STRUCTURE OF METALS. 289
men; the following photographs were taken after the
to 200^ C. for the time shown in the table : —
amber.
Days.
Hours.
21. . .
—
175
12. . .
I
175
3. . .
2
175
4 . . .
5
16
5 . . .
39
20
1 . . .
39
20
icture characteristic of freshlystrained lead, with one
ave persisted from the original crystallisation. In
rrown considerably, and a general change of pattern
g feature is the large skeleton crystal that hafi
/ corner of the marked area. This skeleton is seen
1 figs. 22 and 23. Figs. 24 and 25 were taken under
in order to show another large crystal which gave
ig. 24 it is still somewhat skeletal, but in fig. 25,
3onsolidated, all its outlying arms have disappeared,
lefined crystal, part of which is seen as a dark arm
h of the specimen at the same stage as fig. 25, but
jrs), and so illuminated as to bring the new crystal,
, into brightness. This new crystal is seen to be
irs, and from its position relatively to the marked
ime crystal whose early stages are seen in figs. 22
3lJent example of what may be called an aggressive
3, also at 8 diameters, is shown in fig. 27, Plate 10.
men can be seen, and the photographs illustrate
stals are generally near the edges of the speci
3 Jarge crystals are not mere surface layers, but
ess of the specimen, and can be readily identified
s case, the specimen is a plate about oneeighth
J in the annealed metal is apparently in no way
3,1s in the original state before straining ; the
in a specimen whose original crystals were
by the photographs (figs. 26 and 27) of these
ce of twin crystals, both as inclusions in the
3S. In fig. 26 three distinct sets of straight
2 i>
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rSTALLINE STKUCTURE OF METALS. 291
th oi ciystals occur in lead only when the metal has
)vere plastic strain. The structure of a cast specimen
ures which cause a strained specimen to show rapid
i arranged to cause rapid cooling, specimens of lead
lute crystalline structiu'e, whose scale is not very
Tushed lead ; such a specimen was exposed to 200° C.
sibJe change of structure occurred. A piece of this
severe crushing, and on further exposure to 200° C
specimen is cooled from temperatures of 200° C. to
rtemperature has no visible effect on the structure.
has no visible effect ; quenching in water, cooling in
were all tried on a number of specimens without
3ned to a small extent by severe strain, and the
^storing softness is correspondingly small. In one
^ad was crushed under a given load in the testing
mtil no further creeping occurred. The specimen
i under the same load, when a distinct amount of
lace.
)ed above as having been made with lead were
nt lend themselves to similar treatment ; those
well shown when a surface of a cast ingot of the
iJoric acid. These crystals are generally large,
s obtained on etching the surface of commercial
ntercrystalline boundaries may be seen on the
e grooves or channels. The presence of these
16 method of manufacture, during which these
ted tin, and allowed to drain. Ab the plate is
it crystallises, but any fusible impurity present
' longer, and, being forced by the crystallising
s, the still fluid impurities will drain off, thus
i of comnciercial tinplate is shown in fig. 28,
lalf the natural size. In this photograph the
irly seen, but it also illustrates another and
ixll ca^es of an etched crystalline metal viewed
v^ed that, under a given incidence of light,
^were dark, and that the illumination was
;tal. In the etched tinplate this is not the
l» 2
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TALLINE STRUCTURE OF METALS. 293
f of true crystalline plates ; such distortion would
in the coefficients of expansion of iron and tin
rate of cooling. Considering the extreme thinness
t of distortion might well be purely elastic and
crystals of tin.
specimen, but to the same scale as fig. 31, illustrates
and arrangements of the tin crystals that can be
^ solidification ; the crystals in fig. 31 were formed
>se in fig. 32 by quenching the specimen in water
^as still melted. By means of local quenching and
attems can be obtained ; such processes have long
nufacture of what is called *' moirSe metallique."
it the small crystals of tin which are obtained by
water do not show any growth when the metal is
?mperatures short of the meltingpoint. Even a
does not make them grow or rearrange themselves,
a', be reduced to a minutely crystalline structure by
lens so treated we have observed recrystallisation to
ts on the recrystallisation of cadmium at moderate
can be strainec^ by compressiou until its crystalline
gh interpenetration of the original larger crystals.
(12 diameters) photograph of an etched and marked
strained piece of cadmium. Fig. 34 shows the same
exposure to 200® C. It now shows a welldefined
hows the same area again, after six days' further
considerable increase in the size of the crystals is
;he gradual growth of some of the crystals is very
itures that we have observed in the case of lead are
1 we can see no invading branches and no aggressive
;o be any considerable amount of twinning,
just described were also made on specimens of zinc,
* zinc strained by compression at ordinary tempera
on exposure to 200*^ C. Some results obtained with
electric batteries, were particularly interesting. It
lechanical properties of zinc are widely different at
ly that the metal is soft and ductile at temperatures
it is generally worked at that temperature, while it
at and above 200'' C. Commercial sheetzinc, rolled
emains fairly soft and flexible at ordinary tempera
e is too minute to be seen in specimens etched with
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TALLINE STRUCTURE OF METALS. 295
has been very widely believed that annealing or
iron and steel, are "critical" phenomena which can
^finite temperatures. Arnold has gone so far as to
such an " annealing point." Various of the " arrest
and steel have also been regarded as representing
Baling, but the connection between the two is by no
uud phenomena of annealing or recrystallisation in
e interesting ^o inquire whether any corresponding
the cooling of these metals. We investigated the
trie arrangement consisting of two thermoelectric
ionval galvanometer, and a potentiometer somewhat
OBERTS Austen ; the deflections of the galvanometer
us of a telescope and scale, instead of being photo
)e, therefore, that either from this cause, or from
whole arrangement, some minute arrestpoints were
Itingpoints and the ordinary temperature of the air
observed in the three metals tried, i.e., lead, tin and
f they exist at all, may be found at much lower
1 our experiments were carried,
te the phenomena of recrystallisation in lead, &c.,
1 heat is evolved during the cooling of the metal,
xt even in iron the arrestpoints are not necessarily
ing, we look for a theoretical explanation of these
he theory of recrystallisation which we shall now
s for the explanation of the phenomena described
lit part in the action to the impurities present even
ies which we believe to be of importance are those
utectic alloys, or fusible compounds, with the metal
nainly metals, particularly the more fusible metals,
, mercury, sodium, or even rarer metals, such as
when a metal containing a small proportion of such
[ties are, for the most part, segregated in the inter
•^stals themselves form at a temperature when the
lid, and the growing crystals gradually push the
laries. Where the quantity of impurities present is
n be seen under the microscope forming an inter
where the " pearlite " plays the part of a eutectic,
^ucture ; other examples can be found in the gold
^ Messrs. Heyoock and Neville.* Where the
very small, the meshes of intercrystalline cement
YCOCK and Neville, *Phil. Trans.,' A, vol. 194, plates 4, 5.
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^ALIJNE STPvUCTURE OF METALS. 297
\9 over different crystals, in such a way as to produce
een adjacent crystals. Another phenomenon, seen
ic acid, is also of interest in this connection ; it has
esent paper (see p. 284). We there have a case of
and deposited upon another crystal in its proper
ik it must be admitted that different crystal faces,
heir elements, differ in solubility in the same solvent.
difference is a further step in speculation which is,
connection. Such differential actions may, however,
differences of electrical potential in the surfaces
w of the matter, then the diffusion across films of
olysis. Now, while diffusion in metals and alloys is
ectrolysis in an alloy has not yet been demonstrated
land, the close analogy with salt solutions leads one
ctrolysed, and those who have experimented in the
rtain that greater experimental resources will not
5 phenomena of recrystallisation which the solution
oes not cover, while the electrolytic theory explains
> fact that only strained crystals will grow, while
ency to change even at higher temperatures. The
eory, is that in the unstrained state the crystals are
uous films of eutectic, and that electrolysis only
itortion has broken through these films in places,
ne into contact ; the electrolytic circuit would then
:>ne crystal to the other by direct contact and back
*ecrystallisation in solid metals may be summed up
I is one of solution and diffusion of the pure metal
> fusible and mobile eutectic forming the inter
results in the growth of one crystal at the expense
n solubility of the crystal faces on opposite sides of
bable that this phenomenon of directed diffusion is
:iKR (*Comptes Eendus de TAcad^mie des Sciences,' vol. 116,
Dii into iron is affected by the action of an electric current. He
► iron electrodes enclosed in a fireclay tube ; the whole was
iperes was passed for three hours, when the anode was found
idorgone considerable cementation. This action in the interior
carboniron eutectic.
2 Q
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ALLINE STRUCTURE OF METALS. 299
of the weld and there ended quite abruptly. It
that this weld line was mechanically weak; it
icult to cut or tear the metal along the weld as in
erefore behaved as a true intercrystalline boundary,
tectic, and therefore forming a barrier to crystalline
earance of such a weld in section after annealing
30 diameters. The line AB is the weld. As these
Qiercial lead, we were prepared to find that, as a
eutectic would have occasionally found its way into
seems to have happened only very rarely. We
nd only in two instances did we see a slight amount
1 line of the weld. We think that we are justified
IS to the accidental presence of impurity.
If we have in a welding surface an intercrystalline
crystalline growth owing to the absence of eutectic,
)lied, growth should occur there as elsewhere,
erpose a thin but continuous layer of leadbismuth
in welding ; the specimen was then annealed for
above the meltingpoint of the eutectic — but on
the layer of eutectic had persisted as such, and
lit in this case the film of eutectic introduced at the
litions were therefore analogous to those which hold
ystals, where, as we have pointed out, growth does
nent conclusive it was necessary to have a discon
weld. We accordingly tried another experiment,
3S of the same alloy, and after aimealing we found
►ssed the line of the weld in many places. This
times, various impurities being used, such as the
c, pure tin, cadmium, bismuth, and mercury. All
)wing considerable growth across the weld after
t the amount of growth observed varied very much.
3 of crystals that have grown across the weld ; the
iicated by a discontinuous line, CD, probably repre
i nonmetallic character, around which the crystals
ound the slag in wroughtiron.
I to the action of the impurities which were intro
ite the possible contention that their action was
of the nature of that of the " dirt " more or less
ns, certain further experiments on welds in lead
introduced at the weld was —
2 Q 2
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TALLINE STRUCTURE OF METALS. 301
of a single specimen of freshly crushed cadmium,
continued exposure to 200° C, under oblique light,
• 12 diameters. (Plate 12.)
sheetzinc by exposure to 200^^ C, oblique light,
ers. (Plate 13.)
I weld in lead, using clean surfaces, after prolonged
r the weld is seen at AB. Oblique light, magnifica
late 13.)
I weld, using eutectic in the weld, after prolonged
>f the weld is at CD. Oblique light, magnification
13.)
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Phil. Trans., A.^ Vol. igs, Plate 4.
Etched sheetlead, x 100.
Fig. 7. — Freshly crushed lead, x 30.
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Phil. Trans. ^ A.^ Vol. ig^^ Plate 5.
Fig. 9. — Same after six days.
\
Fig. 11. — Same after two months.
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Phil. Trans.^ A.^ Vol igs^ Plate 6r
Fig 13. — Same after six months.
\
Fig. 15.— Same sheet lead. X 12.
30 minutes at 200** C.
After
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■> c.
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Phil. Trans. ^ A.^ Vol. igs^ Plate 9.
Fig. 25.— Same after 40 days at 200^ C.
is; Fig. 25 (after 40 days at 200^
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1 ^s n
Phil Tram., A.^ Vol igs^ Pl<^te 10.
nnealiog. X 8.
Pig. 29.— Tinplate, etched. X 100.
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Phil Trans., J., Vol igs, Plate 11,
I. X 100.
Fig. 32.— Tinplate, after
remelting the tin and cooling
quickly, x i.
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Phil Trans.. A., Vol. igs, Plate 12
X 12.
35. — S>ime after seven dajs at 200° C.
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Phil Trans,, A,, Vol igs, Plate 13.
0" a X 8.
D
g. 38. — Weld in Lead, with eutectic. x 30.
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03 ]
I in a Magnetic Field.
D,y F.R.S., and Alfred Hay, B.Sc.
Read June 21, 1900.
1421.]
jrimental and partly mathematicaF, has
isional cases of magnetic lines of force
low of a viscous liquid.* The original
ms made, showed that the stream lines
estion, gave results very similar to those
r the cases of an elliptical and circular
at the stream lines imder these circum
direction of the corresponding magnetic
ould be used for many practical investi
long research dealing with the various
I extremely laborious, extending without
a
ise some method by which a thin sheet
n could be obtained of any required
veen two sheets of glass, the required
e formed.
w8 connecting the thickness of the thin
rough it in a given time, so that the
g to the differences of permeability of
pertained.
s undertaken of some cases suitably
evere a test as possible for ascertaining
i for any case, accurately, the position
Ic field.
ling in succession a very large number
.n ellipse with the major axis parallel
Lssociation Report ' (Section A), Bristol Meeting,
20.12.1900
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^ LN A MAGNETIC FIELD. 305
iracteiises the method, is the fact that it
the lilies of induction not only in air, but
tei'ial itself. This cannot be accomplished
1 has therefore been applied to determine
for a number of cases of mathematical
plotted by calculation. Inasmuch as the
f two liquid films with the corresponding
bove investigation been determined, it is
e which may be of interest, and some
)f the method in cases of interest to
icli has been necessary, has not — as far as
tble in any published form, and although,
it, parts of the problem have been dealt
le account given will be found of use in
of the relation between the thickness of
e of flow.
(Oil with theory.
)btaining the streamline diagrams.
out mathematically and plotted, and its
>tograph obtained by experiment,
athematical interest, and also some of
;trical engineer.
jstigation of the subject, and
I elliptic cylinders and confocal elliptic
er A, tig. 1, from which it is forced by
etermined by the pressure gauge B. It
nt being secured by the lever D which
E, ^Nvhere its temperature is recorded
he slide J.
ss fixed within the frame P ; a well is
is introduced, a channel being formed
Eii\ The correct thickness is obtained
less gauges N, N, N, N, placed at the
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IX A MAGNETIC FIETJ).
307
Film between Parallel Plates.
20 lbs. pressure.
wing through in 10 minutes.
to flow,
minute.
B.
c.
> 19
miuutes.
527
24
. 616
162
518
110
91
695
168
595
942
1933
518
975
26875
37 3
1255
429
234
1665
557
18
210
861
116
271
D786
928
320
172
804
329
!49
605
416
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IN A 1
^tAGNETtC FIELD.
f
;
/
J
t
i
/
•
J
I
J
/
\
/
1
^
/
*>
/
/
^.
■
^
■—
B^
ILm in /V7C>
n quduitii
ftlNE
7
has.
e.
•O/
5
I
■
A
1
1
/
S
/
1
/'
1
/
}
/
1
i
/
J
J
r
7
^

■>.
B_
in .
I/Ve
fnch
n b
4dLn
imt
'i
><0
809
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S^ A MAGNETIC FIELD. 311
en X — ^ty the velocity at the boundary
)er unit width ot the layer of thickness
2 vdx
Jo
lin layer between parallel plane walls is
^er, seen to be proportional to the cube
experimentally shows a satisfactory
able us to calculate approximately the
not, however, well adapted for exact
it difficulty in measuring accurately the
f the glass plates are either sufficiently
eir great thickness) for refined measure
ibe of the thickness of the liquid layer,
xtion of t win give rise to a large error
1 for the coefficient of viscosity in C.G.S.
nding to a thickness of '012") the value
and which may be accounted for either
Y slight irregularities in the containing
tre — all of which causes combined might
:' the viscosity equal to 2*5 ; the density
in fair agreement with the results of
out careful absolute measurements, but
to apply the method to twodimensional
! curves obtained (figs. 2 and 3) were
these experiments whs ordinary tap water, which
[cation and Lubricants.*
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IN A MAGKETIC FIELD.
313
4.
Iflmpi
3
ew.
D.
view.
O
}
^lates and Clips.
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A MAGNETIC FIELD.
315
►vas taken to a place where it could be
ire given to the sensitive photographic
Zinc templates of the shapes required
the wax, the outline was cut with a
med. In cases where the flow in the
)ned, and could only be removed after
s will be seen in the photographs, figs,
^ed at all.
I.
jtreamline method to the solution of
decided to work out mathematically
ction placed in an originally uniform
3 field, to plot the lines of magnetic
Moo, and to compare the diagram
theoretical diagram is given in fig. 9
agram in fig. 10. In making the com
10 was prepared, and fig. 9,"**' which
s then superposed on it : the coinci
)rily established the soundness of the
\ that slight local divergences along
stead of the sharp refraction of the
leoretical diagram, we have in fig. 10
perfectly straight lines crossing the
:• or smaller extent, in all the stream
It is more marked in those cases
the two liquid layers, i.e., the perme
, is greater. It is clear that the
l)s the originally uniform distribution
comparison between the theoretical
ary to assign to the liquid layer an
hat of a particular streamline in the
lid in fig. 10 is clearly show^n by the
tlie shape of their boundaries is the
The method by which the solution
[ fully explained in the mathematical
boundaries of the liquid layer are
means that the diagram gives the
tion of this paper.
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IN A MAGNETIC FIELD. 317
e this latter streamline diagram with the
3nt in the shape and general distribution
triking.
im for an elliptic cylinder of permeability
iced in a uniform field with the major axis
:ing an angle of 45° with the impressed
for a cylinder of the same permeability,
. 23 is a similar diagram for a very thin
How cylinder of fig. 16, but here turned
ed that fig. 24 confirms the theoretical
lollow elliptic cylinder bounded by two
field be uniform.
I may be treated theoretically as well as
ses is very limited, and the vast majority
beyond the powers of analysis. It is in
ployed by us becomes a powerful weapon
s of rectangular section, figs. 25 and 26
* square section placed with one of their
o it respectively. In fig. 25, where the
»ng the direction of the impressed field,
ir are concave outwards ; in fig. 26, on
a,pidly as we proceed along the field, the
outwards. This suggests that an inter
x)und for which the lines exhibit neither
Lnd, as a matter of fact, we know of one
r. Fig. 26 closely corresponds to the
cently succeeded in working out analyti
tiduction.* Figs. 27 and 28 are intended
bh of one of the sides of the rectangular
We know from theoretical considerations
magnetising factor, and thus increasing
very clearly brought out by a comparison
3tic fields corresponding to a cylinder of
inside a hollow square one, and a solid
teresting in connection with the problem
s are slightly disfigured by airbubbles,
^f, once they are allowed to reach a part
film varies,
ry, 1900, p. 225.
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[ A MAGNETIC FIELD. 319
^iical Appendix.
'ptic Cylinders and Confocal Elliptic
tical theory ot magnetic induction was
was the first to work out in detail the
nagnetic material placed in a field of
id his investigations to the case of an
it a later date, gave an approximate
nth its axis along a uniform field. In
1 ellipsoid of revolution placed in any
I in solving the same problem for a
A. G. Greenhill considered the case
72,\\ diagrams of lines of induction for
erial placed in a uniform field. On
;ure in almost every textbook on the
nown. In Maxwell's great treatise
iagrams. These include the following
•netised transversely, and placed with
)ther (vol. 2, fig. 14); (2) a circular
5e direction, and placed in a uniform
;he cylinder is coincident with that of
al in a uniform field (vol. 2, fig. 15) ;
uniform field whose direction is at
1 of the cylinder (vol. 2, fig. 16) ;
in infinitely long straight cylindrical
ol. 2, fig. 17).
iling with the induced magnetisation
[e obtains a solution by assuming the
B/r) cos (f>y where r is the distance of
^r, and (f> the angle between r and
tants A and B assuming different
substance, and outside the cylinder
^7'^ is an integral of the equation
function Az + B^ of the complex
t Ibid,, p. 66.
§ * Journal de Physique ' (1881).
! Magnetism,' pp. 493495.
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IN A MAGNETIC FIELD. 321
)s of the two media, and itj, Wn for the
into the coiTesponding media,
drical shells, it is convenient to abandon
s, and to have recourse to the circular
=1 (1)
> — b^, represents the family of ellipses
V
r' > J/ > — a^, represents the family of
= sin 6,
~ At
f,  — =sinhw,
[b + I/)
ellipse, and = constant to a hyper
t(> oo ), and (which may vary from
>osition of a point in the plane of the
nates.
I potential function, we make use of
^11 that the magnetisation of a solid
et one of the axes of the ellipsoid be
lensional case of the ellipsoid to the
d we see that in this case also the
might be supposed to be produced
inary magnetic matter, of volume
nt, to be displaced relatively to each
I of magnetisation, such that pBs is
the cylinder. If then pY stand for
cylinders of the imaginary magnetic
9V
isplaced cylinders is p ^ Bs. But
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N A MAGNETIC FIELD. 323
v^arddrawn normal at any point of the
L becomes
av.
putting u = tanhi  we get for the
(m1)'>H (3).
.H.
f. COS — (fi ^ l)ah sinh u cos 0}
d from these equations.
le cylinder is obviously ij = constant,
il lines, we have to determine the
ce the function which is conjugate
te to sinh u cos is cosh u sin 6,
— T — sinh?4
l)ao
, K being a constant which varies
nipressed field is along the minor
>ii to the lines of the external field
 , Sinn ?/,
le line to another.
elliptic cylinder may be extended
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IN A MAGNETIC FIELD. 325
n w = u^, Vi = V^ ; this gives
t + fcB, . (4)
A^ cosh t^i — Atf sinh Uj (5).
Si( ^ Bit '
. — jxbA^ — /jtaBj . . .... (6),
I — fiA^ sinh u^ — /aB cosh i^j . . . (7)
A, Ai, B^, and A^, we get
^cosh^i) , — im
 l)a6(/Lico8hi^ + sinhi^^) * ^^
sh 2c^ 4 sinh Ui) , i \ TT
l)aJ(yt6C0sh?<i + sinhwi) ' ^^
a somewhat simpler form by putting
t^y 6i stand for the semiaxes of the
ering that u^ = tanh *" ^ (&i/«i), we get
TT • H
^a7}(^)
— Oft)
 «i6)
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N A MAGNETIC FIELD, 327
space,
the substance of the shell, and
external space.
the view of plotting curves, it is con
! cosines, such as the one compileil by
?al Society of London. Corresponding
I easy step to pass to Cartesian rect
it for plotting the r^urves.
mentioned, be extended to any number
il, however, wlien numerical data are
>uce in the equations instead of first
lis may be regarded as a limiting case
^coming equal. If in the expressions
e case of a hollow elliptic shell we
II proceed to the limit h = a, we find
i\'ith those deduced by Du Bois for a
netic Shielding."* But although an
cial case is thus obtained, it is much
lells, to follow the method developed
S Seddon 2, p. 613.
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PhiL Trans,, A,, Vol, 195, Plate 14.
10
imline Diagram corresponding to
iheoretical Diagram of Fig. 9.
12
inite Elliptic Cylinder in uniform field,
^atio of axes 3:1. Permeability 1000.
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Trans,, A., Vol. ig^, Plate 15.
14
Cylinder in uniform field,
ability =100.
16
>tic Cylinder in uniform
ieability = 100.
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il. Trans., A., Vol. \g^, Plate 16.
18
iric Shield enclosing Solid
. Permeability =100.
20
ar Cylindric Shield.
line Diagram,
ability = 100.
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Phil Trans., A., VoL 195, Plate 17.
22
r. Ratio of axes 3:1, major
it 45^ to field. Streamline
. Permeability =100.
'24:
iptic Cylinder.
lined 45° to field.
>ility=^100.
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Trans., A,, Vol. 195, Plate 18.
of Square Section.
lity=100.
Rectangular Section.
lity=100.
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Trans., A., VoL 195, Plate 19.
30
of Triangular Section,
ability =100.
3:i
blinder enclosing Solid
Permeability = 100.
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'ns.,A., Vol, \g^y Plate 20.
) and Teeth of Toothed 
Permeability =100.
6
Lture by Polepiece.
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PhiL Trans., A., Vol. 195, Plate 21.
31)
of Magnetic Field near edge of
ole piece in Dynamo with
Toothed core Armature.
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\egrals to Optical Problems.
\ Isaac Netvton Student in the
Ige.
HOMSON, F,RS.
, 1899.
Page
notions 331
>red 331
..,.•.... 331
......... 332
333
334
335
ar a length of time which
)ing detected .... 335
»e estimated by summing
to which it is resolved
335
theory of widening of
vibrations, Thomson's
338
by Lord Eayleigh. . 339
3f the pulses .... 339
o light of composition
i pulse, subject to the
the sDoiallest observable
waves 339
itical expression gives
342
342
cZ hy a law 343
24.12.1900.
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J. PROBLEMS. 331
Page
360
>chromatic light will arouse
reasing intensity .... 361
[Kr's integral 361
some progress with the mathe
mpose natural light,
e phenomena forbid us to regard
iin of simple waves, such as may
ne, the equations of optics find
iesirable to enquire how far we
)f simple wavetrains by means
procedure was first suggested by
permissibility of this process,
and independence of the simple
be found an attempt at a strict
ciples (i.) that we are cognisant
luced by the light during an
the detector in use (the eye, a
ned with simple wavelengths,
integrated energy we observe,
as been put forward with great
md planepolarised light.
:ter than by a quotation £rom
srplications qu'elle donne des
vement simple, dans lequel la
• una Equation de la forme
cements of an elastic medium, or of
sctors, which can be interpreted in
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:jal problems. 333
efining it for a finite interval alone.
les of the time, from — oo to + oo.
in nt)du
fiy) sin uvch\
a
f elementary simple vibrations, of
Ht)
results of the undulatory theory
all values from zero to infinity,
nber of simple circular functions
have meaning, as in the familiar
nstruments, &c. The question we
g in the limit, when their number
1 ; this was, in fact, offered by
ach of the component vibrations
all time. This is true whatever
ng. But this disturbance may,
? interval of time. Take the case
I. PoiNCAR^, will separate the
ed separately. Hence a spectro
time before it is kindled, and for
Is must therefore be fallacious.
spectroscope possesses infinite
telescope will be illuminated by
le matter from another point of
rruji'ing in Physical Problems. — In pure
Is subject to certain limitations. These
luities, or infinite sets of fluctuations, or
d neither infinities nor discontinuities.
commonly used to represent physical
t character of the method ; a function
cjucstion would be without infinite or
95 ; also see Schuster, *C. R./ 120,
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AL PEOBLEMS.
335
K  ^sM^
3 upon the distribution of energy
■ phases of corresponding elements
order to prove that, in case of a
e amount of interference depends
any assumption respecting the
onsequences.
•nfined to the particular case of
perceptible fluctuations or other
I the long waves of Hertz is by
jreat compared with the periods
sed to perceive and register the
r chemical effect, photographic or
minescence which the radiation
le features of a single wave can
ht, with a view to discriminating
lely with the integral effect over
isider the molecule as a simple
istified, we may prove that the
ling but the partition of energy
:er integral. The phases will of
it
id.
or no ; but, if that condition
+•
i/(«) depends upon ^f{t)dt
— 00
qual to
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AL PROBLEMS. 337
'erent periods drop out (a familiar
ite of absolution is dependent on
►rptiou. So again with physiolo
s produced by light we probably
le of luminescence also. It may
* dissociation. But, if the disso
excited vibrations become large,
ihtle88 some molecules will split
iile the precise timing of its own
be allimportant. But on the
f dissociation will perhaps depend
ifying the average structure,
linear equation for the vibration
3t step towards a solution of a
>KES, ''Linearity applies to the
} ether — but it does not follow
omplex system of molecules."
t treatment to the spectroscopic
en direction is compounded of
e of the instrument, and the
^ tff — depends upon the same
ip — depends upon i/r..
rznining the constancy of the
t is constant, so also is the
iiy ; hence, the spectroscopic
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PROBLEMS. 339
urve to completely specify the
ts will be justified below.
.EiGii* in his paper on "The
proposes to regard this as an
(8)
(22)
% is therefore C^e'^'^^du.
necessarily equal to (8), and of
le range from — oo to + cx), the
value of u would be indeter
could be assigned ; and if the
iltant would again be indeter
; there is room for an infinite
ioncerned only with an average,
to be. proportional to the total
phaserelations. In the aggre
^y is distributed is still, for all
theory. But when we come to
are certain questions which still
is confined to a certain small
ssor Thomson's theory, how
;ive the properties which Lord
Again, we might suppose that
egrees of crowding among the
b a great number of them are
)e so thinly scattered as to be,
occupied by each. Experiment
1 correct ; we may enquire what
rd Rayleigh's theorem to the
us, be equivalent to a spectrum,
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PROBLEMS. 341
base of the resultant will be
c motion A.
lall, the new phases will differ
leat, the new phases will differ
11 have no apparent connection
radiation. First, we can only
)h we have taken to be T ; we
pulses in T. Now Schuster's
dvgj of a radiation, and its ex
iIer expression for the resultant
ion of V, but partakes of the
this point we make use of
1 particular wavelengths, but
avelength. Bearing in mind
>es, we have a spectrum whose
Lirve will be less rapid as we
)ave just seen, when the time
ms are so crowded as to be
the meancurve n(f}%u). But if
e of angle included in the set
se intermediate between the
e pass continuously to quicker
) and so on ; the divergences
t on a broad and theoretically
miount of energy in the slow
)f Sound,' ed. 1894, p. 40.
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niOBLEMS. 343
L much lai'ger exposure, or as
are not all similar is obvious.
es of constant displacement^
inie time the proportions of
tween x and x + dx are, say,
ulses ; suppose that they give
* the mixture is
jight.
)n by supposing it to consist
is practically a pulse of con
thickness of a pulse is com
r the cathode stream. Lord
»e regarded as simple waves
)v the properties of a succes
mcy of statement. Professor
new. He has held that the
Now this is a valid objection
rht of any composition what
sess definite phaserelation ;
)f the present essay that the
truna of definite composition,
of being brief in comparison
1 be a great number of them
iverages over small ranges of
es of time and wavelength
cession of pulses and that
3he single pulse.
1 the scale of Avavelength.
of X which is zero from
CO
y, 1898.
•, 1898.
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>ROBLEMS. 345
ignetic force great and nega
legative are to be so balanced
pulse shall vanish. On this
IS, Sir George Stokes* bases
be zero amplitude for the
lergy in the visible spectrum
wfU be of order E^d . {d/X^y,
stead of 10"^. Now diffrac
j order ; very much shorter
and in any case would give
posed forms will be sensibly
3h higher degree than those
IS,
nber of independent pulses
an incandescent gas. We
I, and to have no visible
tion as composed of plane
Furthermore, the amount
ected by absorption. The
free vibration,
omogeneous. One reason
les in the line of sight will
doubtless be the altered
This will perhaps become
r it at present,
iie train of single waves
gth, but has a definite
ed below. The Doppler
arrive at the remarkable
3 pressure is indefinitely
ted by Lord Rayleigh,
ion virith another. The
3ction. The vibration
>re be suddenly and
otal radiation received
)7.
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PROBLEMS. 347
ig off from a maximum at k
(i)
), where 7i is the distance from
avelength.
cs both Athwart and in Line
imum brightness, and definite
erent lengths of train.
v^ith velocity v, a fraction C^'*
'Jr"4
./p1
("•).
Jume,
3ity Vy a fraction
of waves of length between
crht.
Lction ^e ^ dr^ will give free
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L PROBLEMS. 349
hf^'^mi
igh the breadths of elementary
•tly, yet the application of his
yte detailed information. The
retardation u of the two half
eir " visibility " estimated for
isibilitycurve thus constructed
m line, and find out something
intensity of light for position x
idifference.
nd 34.
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PROBLEMS.
351
ble. We axe enabled, however,
ves in his paper on the kinetic
,p«
<
4 1) e^dP
notation.
e values in column 6 of Tait's
^ the function is very near to
•
1 1
1 1 i 1 1
1 —
1
I 1 1 1 i
r
^.
\^
^^
s.
^.:^
^;>v
'"'••.,
=^
^
£0
£•3
>se /x so that the two curves
; graphically we see that we
■^z/^2 > II. (dotted curve) is
he integral through using 11.
s of the order of the errors
t^pare our results.
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)BLEMS. 353
esent paper necessitates a
tted from Lord Eayleigh's
abovementioned necessity
theory. The conclusion of
its for a certain fraction of
rom f to ^ for the different
iby
I
the spectrum on a scale of
11 be Sq. Hence the " half
onnected by the relation
to the visibility halfwidth
should be diminished by a
md collisions comparatively
LS would disappear, the free
:pected that the formula for
rd Rayleigh, in which the
g is valid, there is no such
indefinitely reduced; the
It is noteworthy that the
oes Lord Rayleigh. This
tation.
lathematically homogeneous
certain continuous spectrum,
yrsis that, for zero pressure,
r curve for the total spectral
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)BLEMS. 355
iomponent curves are, the
lUow ourselves to substitute
these become narrow,
ied for r, will not affect the
th of train is taken to be
ater velocities give shorter
)ther ; for it gives too great
dties, which velocities send
of the error is to make the
he limiting width for zero
, and then to proceed to the
me.
rum Lines.
3 vibrations of an atom are
umple gas will be to some
loped by JAUMANN.t
ibration
(x.)
IS been pointed out in the
) physical meaning for the
lent emissions of this form
* a gas cannot strictly be
(x.) will never be allowed
a stage by a new collision.
)f a naolecule has generally
shall not be making a
ives. We will proceed to
^s so rapid as to allow this
application of Fourier's
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OBLEMS. 357
mogeneous Light
oducing a large number of
f a certain regularity in the
las been abundantly refuted
)t be produced without the
iges is an index, not of the
)f the spectroscope,
out a spectroscope, by using
nogeneity. The number of
case, it is also a test of the
y a Fourier integral of the
ill fluctuate rapidly in terms
uency p^ we will use the
values of u small compared
ut + \lf)du.
to a region on either side of
 \lj)du
with those of oospt. The
Biiote a simple vibration of
of the expression will be,
[)ply to
e Theory/' « Encycl. Brit.'; ' Phil.
^94
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MS.
359
ot is of the same order :
=i. In accordance with
3^ wjwelenjjths ; while
Hjueiicy of the light
a naturtil nxdiatioii {see
:er of this ti'eatmeiit of
has a historiciU inteivst,
may he worth while to
vibrator is 8oon to dopond
The effect of tlie irrogii
^ning of the Rpectruni line,
svhich laid the foundatioiiH
TU) natural radiation ih a
t of the irn^gularitit^H upon
it of tlie natural vibratiouH
ukritieH in the light would
K 520.
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[S. 361
iiiiig of the incident
11 of a large number
of the amplitudes,
nil of the component
iiergy of the natural
•ns ; in other words,
will become greater
there is no tendency
interact one another.
jes are, so to speak,
IS light will give a
e^being entirely due
1 be prevented from
ponent trains would
ation settles down to
5 irregularities,
plication of Fourier.
:egrand involves an
re that the integral
I We are forced to
s to the actual pro
he natural vibration
the observed effects
•n JR^rfu will excite
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t/ Marble.
rqfessor of Gex>logy in
M.Inst.C.E., Head of
lical Si'fiool {foi^ierly
y).
R.S.
Page
363
369
370
370
373
376
jr . . . . , 382
tificial Deforma
386
tificial Deforma
lighly contorted
387
398
I in many parts of the
manner is a fact which
a glance at any of the
ich have been prepared
the hardest rocks have
' or '* flow " of material
e facts are undisputed,
lug, has taken place is a
5.1.1001
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V OF MARBLE. 367
calcite, however, wais
which existed in the
ind cracks, as well as
a finely pulverulent
portions of the mass,
jsure. GuMBEL con
;ticity on the part of
[)f the alahaster, and
In the field, that the
re they had become
hows distinctly under
to a recementation of
*. It was shown by
e in the case of the
d to powder; in the
^ed that deformation
for the phenomenon
jliaracters and optical
I been submitted to
)ut by PFAFF.t He
lofen in a steel block,
V n upon it. A very
estone, and this was
e amounting to 9970
as not displaced, and
specimen of the same
ure of 21,800 atmo
le limestone did not
left on the polished
the conclusion that
y different materials,
5uts reproduces very
; described ; but the
red out to receive a
• the hole a steel die,
ire,
■ Zeit. des Vereines
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OF MAEBLE. 369
Jans of explosives.
Tmmeter of about
od together again
chamber in which
pJetely perforated
' the influence of
what plastic, the
e when removed,
e marble had lost
says, it was seen
into a solid mass.
)und the central
of rocks (and it
iJ work in which
reared up to the
3 and in certain
taken in any of
applied, of the
w cases, of the
5cription of the
strength of the
:ie pressure can
ing about the
s crust, where
ssion will not
ce molecular
niovement to
assure affects
k ^virkt, der
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OF MAEBLE. 371
mit combined with
1 however it was
^ tubes of wrought
nanoe by rolling a
le strip to the bar
is then bored out,
uarter of an inch
the tube instead
he requirements
) of the marble,
in length, were
ff, of Gottingen.
ajs accomplished
he tube, and so
tube when cold.
pass completely
)n allowing the
aed, and it waa
3red indispens
; applied, as it
le experiments
>r immediately
an accurately
3 was applied,
ising a double
accompanying
ist of square
)osite to it by
5 marble with
being kept in
to cylindrical ,
them when
in diameter,
las its upper
3tion to the
rrosion and
[na are cen
sures to the
pper end of
lure is kept
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W OF MARBLE 873
ly bulge. This bulge
rounding the marble.
igns of rupture, when
rratures.
at ordinary tempera
Perent cases, the con
ns taking place more
ninutes to 64 days,
ceased, and in this
The final amount of
ved signs of rupture
a close.
imn before the pres
) completion of the
y slowly i the time
ble, consistent with
;, and the tube was
ling machine along
as found to be still
5, now completely
. without mechani
in between them,
itting the marble
,. tube adhering to
1 the former case
vhile in the latter
or two instances
le latter a smart
^ that it could be
e exterior surface
;h and conformed
all the fine tool
y from the tube
r the dimensions
with equal case
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V OF MARBLE. 873
Y bulge. This bulge
'ounding the marble,
gns of rupture, when
ratures.
it ordinary tempera
erent cases, the con
s taking place more
linutes to 64 days,
ceased, and in this
The final amount of
red signs of rupture
3 a close.
mn before the pres
completion of the
y slowly, the time
Je, consistent with
, and the tube was
ling machine along
3ts found to be still
I, now completely
without mechani
in between them,
Itting the niarble
tube adhering to
1 the former case
^hile in the latter
or two instances
^e latter a smart
y that it could be
e exterior surface
th and conformed
. aU the fine tool
y from the tube
f the dimensions
, ^ith equal ease
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y OF MARBLE. 375
extent by the cones
as above mentioned,
h the unaltered cone
en examined under
h had taken plia.ce.
r its turbid appear
aic of the unaltered
in Experiment P of
ved and branching
through the rock.
)andsof very small
place. The calcite
so produced have
fter the movement
mbers of irregular
have been carried
The structure is
Ispars and many
ble showing this
ty and magnified
'067 inch, which
it bulging. The
nulated portions
itinctly twinned,
movements of
be seen to have
attening of the
have been bent
it cases, whichi
polysynthetio
ihe destruction
I of the calcite
essure to alter
lently due to
111 be referred
of somewhaj;
this structure
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OF MARBLE, ^77
ice (E). The whole
Bunsen flame. The
learlv {\s possible at
le extreme limits i>f
temperature
!y hiH jKwgeiblo
inn ill inches
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OF MARBLE. 381
ut unfortunately it
which rendered it
n contact with the
around the central
'here had been no
ered marble tested
When sliced and
caclastic structure,
3k a foliation which
ed the very narrow
scribed. The twin
by strain shadows,
nor very striking,
into very irregular
irms (fig. 6), quite
The individual
lounced movement
* twin lamellae, is
3
the calcite indi
ke outline of the
twin lamellfe. In
I inward between
1 interest, as it is
over one another
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F MARBLE. 383
•med while at a
ier a pressure of
slowly and at as
2 mouths. The
Uiushing load after
deformation.
longer than the ori
ginal rock
o 25*51 per cent,
in former experi
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W MARBLE. 385
leformation is not
1 and some trans
ous mosaic before
li section, starting
>ward the middle,
ones. Under the
amber of fine and
ilong which there
vse, and elsewhere
aclastic structure.
)n may be said to
e, accompanied by
seen in the case
0^ C. The calcite
al (none are more
rmed rock a very
flattening of the
long as they are
but no twinning,
aning, givbg rise
will show strain
c twinning at the
a when magnified
lamellae in several
of between '0005
I rock, it appears
the calcite grains,
above described,
even in this iron
the deformation.
1 of infinitesimal
hus contributing
on, however, are
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OF MARBLE. 387
1 three planes or
Q to accommodate
ids. The action is
sUps, the portion
olid. The process
igate effect is not
Lination in metals
he crystal on one
) need to suppose
dip when it occurs
because the metal
, and is able to be
idividual crystals,
its shape and its
occurring within
h those presented
at the agi'eement
7 applicable to the
Inscribed, as it is of
:al is squeezed flat
ARBLE BY ArTI
LlMESTONES AND
J Crust.
h s crust has been
lalfcentury, com
of limestones and
Hy with unaltered
ing from pressure,
mites from many
sections of these
Neozoic age, does
served, and states
seen, producing a
[>s, one of them a
er K. Bayer. Akad.
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OF MARBLE, l\H\)
a.
la.
^anada.
Janada.
, Canada.
)tful Origin.
SCructureif.
la.
la.
I.
uada.
la.
/anada.
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L
f
k
T
y
a
;d
ELS
of
be
ge
he
st
tits
}tic
%
lOSt
'OW
her
i of
;up
axe
^hat
ight
how
In
from
ding
med
and
rmed
age
ouble
( the
labby
?en to
)arely
jalcite
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f
s
t
s
e
a
a
i
i
i
s
t«
s
3
r
a
1
1
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